data_source stringclasses 9 values | question_number stringlengths 14 17 | answer stringlengths 1 993 | license stringclasses 8 values | data_topic stringclasses 7 values | problem stringlengths 14 1.32k | solution stringlengths 11 1k |
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college_math.PRECALCULUS | exercise.9.2.32 | $-\frac{5809}{990}$ | Creative Commons License | college_math.precalculus | Express the repeating decimal as a fraction of integers: $-5.8 \overline{67}$ | $\boxed{-\frac{5809}{990}}$ |
college_math.PRECALCULUS | exercise.10.7.91 | $\left[-2 \pi,-\frac{5 \pi}{3}\right] \cup\left[-\pi,-\frac{\pi}{3}\right] \cup\left[0, \frac{\pi}{3}\right] \cup\left[\pi, \frac{5 \pi}{3}\right]$ | Creative Commons License | college_math.precalculus | Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-2 \pi \leq x \leq 2 \pi$: $\sin (2 x) \geq \sin (x)$ | $\boxed{\left[-2 \pi,-\frac{5 \pi}{3}\right] \cup\left[-\pi,-\frac{\pi}{3}\right] \cup\left[0, \frac{\pi}{3}\right] \cup\left[\pi, \frac{5 \pi}{3}\right]}$ |
college_math.PRECALCULUS | exercise.10.7.84 | $\left[-\pi,-\frac{\pi}{4}\right] \cup\left(0, \frac{3 \pi}{4}\right]$ | Creative Commons License | college_math.precalculus | Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-\pi \leq x \leq \pi$: $\sin ^{2}(x)<\frac{3}{4}$ | $\boxed{\left[-\pi,-\frac{\pi}{4}\right] \cup\left(0, \frac{3 \pi}{4}\right]}$ |
college_math.PRECALCULUS | exercise.6.1.35 | $\ln \left(e^{5}\right)=5$ | Creative Commons License | college_math.precalculus | Evaluate the expression: $\ln \left(e^{5}\right)$ | $\boxed{\ln \left(e^{5}\right)=5}$ |
college_math.PRECALCULUS | exercise.6.3.10 | $x=-\frac{\ln (2)}{\ln (5)}$ | Creative Commons License | college_math.precalculus | Solve the equation analytically: $5^{-x}=2$ | $\boxed{x=-\frac{\ln (2)}{\ln (5)}}$ |
college_math.PRECALCULUS | exercise.3.4.23 | $i^{15}=\left(i^{4}\right)^{3} \cdot i^{3}=1 \cdot(-i)=-i$ | Creative Commons License | college_math.precalculus | Simplify the given power of $i$: $i^{15}$ | $\boxed{i^{15}=\left(i^{4}\right)^{3} \cdot i^{3}=1 \cdot(-i)=-i}$ |
college_math.PRECALCULUS | exercise.10.7.57 | $x=0, \frac{2 \pi}{7}, \frac{4 \pi}{7}, \frac{6 \pi}{7}, \frac{8 \pi}{7}, \frac{10 \pi}{7}, \frac{12 \pi}{7}, \frac{\pi}{5}, \frac{3 \pi}{5}, \pi, \frac{7 \pi}{5}, \frac{9 \pi}{5}$ | Creative Commons License | college_math.precalculus | Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\sin (6 x)+\sin (x)=0$ | $\boxed{x=0, \frac{2 \pi}{7}, \frac{4 \pi}{7}, \frac{6 \pi}{7}, \frac{8 \pi}{7}, \frac{10 \pi}{7}, \frac{12 \pi}{7}, \frac{\pi}{5}, \frac{3 \pi}{5}, \pi, \frac{7 \pi}{5}, \frac{9 \pi}{5}}$ |
college_math.PRECALCULUS | exercise.6.4.17 | $x=2$ | Creative Commons License | college_math.precalculus | Solve the equation analytically: $\log _{169}(3 x+7)-\log _{169}(5 x-9)=\frac{1}{2}$ | $\boxed{x=2}$ |
college_math.PRECALCULUS | exercise.10.7.40 | $x=\frac{\pi}{6}, \frac{5 \pi}{6}, \frac{7 \pi}{6}, \frac{3 \pi}{2}, \frac{11 \pi}{6}$ | Creative Commons License | college_math.precalculus | Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\csc ^{3}(x)+\csc ^{2}(x)=4 \csc (x)+4$ | $\boxed{x=\frac{\pi}{6}, \frac{5 \pi}{6}, \frac{7 \pi}{6}, \frac{3 \pi}{2}, \frac{11 \pi}{6}}$ |
college_math.PRECALCULUS | exercise.3.4.13 | -10 | Creative Commons License | college_math.precalculus | Simplify the quantity $\sqrt{-25}\sqrt{-4}$ | $\boxed{-10}$ |
college_math.PRECALCULUS | exercise.4.3.4 | $x=-6, x=2$ | Creative Commons License | college_math.precalculus | Solve the rational equation: $\frac{2 x+17}{x+1}=x+5$ | $\boxed{x=-6, x=2}$ |
college_math.PRECALCULUS | exercise.2.3.23 | The rocket reaches its maximum height of 500 feet 10 seconds after lift-off. | Creative Commons License | college_math.precalculus | The height $h$ in feet of a model rocket above the ground $t$ seconds after lift-off is given by $h(t)=-5 t^{2}+100 t$, for $0 \leq t \leq 20$. When does the rocket reach its maximum height above the ground? What is its maximum height? | $\boxed{The rocket reaches its maximum height of 500 feet 10 seconds after lift-off.}$ |
college_math.PRECALCULUS | exercise.2.2.13 | $x=-1, x=0$ or $x=1$ | Creative Commons License | college_math.precalculus | Solve the equation: $|x|=x^{2}$ | $\boxed{x=-1, x=0$ or $x=1}$ |
college_math.PRECALCULUS | exercise.7.3.19 | The bulb should be placed 0.625 centimeters above the vertex of the mirror. (As verified by Carl himself!) | Creative Commons License | college_math.precalculus | The mirror in Carl's flashlight is a paraboloid of revolution. If the mirror is 5 centimeters in diameter and 2.5 centimeters deep, where should the light bulb be placed so it is at the focus of the mirror? | $\boxed{The bulb should be placed 0.625 centimeters above the vertex of the mirror. (As verified by Carl himself!)}$ |
college_math.PRECALCULUS | exercise.6.4.13 | $x=\frac{25}{2}$ | Creative Commons License | college_math.precalculus | Solve the equation analytically: $6-3 \log _{5}(2 x)=0$ | $\boxed{x=\frac{25}{2}}$ |
college_math.PRECALCULUS | exercise.10.7.55 | $x=0, \frac{\pi}{8}, \frac{3 \pi}{8}, \frac{5 \pi}{8}, \frac{7 \pi}{8}, \pi, \frac{9 \pi}{8}, \frac{11 \pi}{8}, \frac{13 \pi}{8}, \frac{15 \pi}{8}$ | Creative Commons License | college_math.precalculus | Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\sin (5 x)=\sin (3 x)$ | $\boxed{x=0, \frac{\pi}{8}, \frac{3 \pi}{8}, \frac{5 \pi}{8}, \frac{7 \pi}{8}, \pi, \frac{9 \pi}{8}, \frac{11 \pi}{8}, \frac{13 \pi}{8}, \frac{15 \pi}{8}}$ |
college_math.PRECALCULUS | exercise.9.1.5 | $x, \frac{x^{2}}{4}, \frac{x^{3}}{9}, \frac{x^{4}}{16}$ | Creative Commons License | college_math.precalculus | Write out the first four terms of the given sequence: $\left\{\frac{x^{n}}{n^{2}}\right\}_{n=1}^{\infty}$ | $\boxed{x, \frac{x^{2}}{4}, \frac{x^{3}}{9}, \frac{x^{4}}{16}}$ |
college_math.PRECALCULUS | exercise.6.3.36 | $(-\infty,-1) \cup(0,1)$ | Creative Commons License | college_math.precalculus | Solve the inequality analytically: $2^{\left(x^{3}-x\right)}<1$ | $\boxed{(-\infty,-1) \cup(0,1)}$ |
college_math.PRECALCULUS | exercise.1.1.28 | $d=\sqrt{83}, M=\left(4 \sqrt{5}, \frac{5 \sqrt{3}}{2}\right)$ | Creative Commons License | college_math.precalculus | Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $(2 \sqrt{45}, \sqrt{12}),(\sqrt{20}, \sqrt{27})$. | $\boxed{d=\sqrt{83}, M=\left(4 \sqrt{5}, \frac{5 \sqrt{3}}{2}\right)}$ |
college_math.PRECALCULUS | exercise.6.1.19 | $\log _{6}\left(\frac{1}{36}\right)=-2$ | Creative Commons License | college_math.precalculus | Evaluate the expression: $\log _{6}\left(\frac{1}{36}\right)$ | $\boxed{\log _{6}\left(\frac{1}{36}\right)=-2}$ |
college_math.PRECALCULUS | exercise.10.7.33 | $x=0, \pi, \frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}$ | Creative Commons License | college_math.precalculus | Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\sin (2 x)=\tan (x)$ | $\boxed{x=0, \pi, \frac{\pi}{4}, \frac{3 \pi}{4}, \frac{5 \pi}{4}, \frac{7 \pi}{4}}$ |
college_math.PRECALCULUS | exercise.10.1.41 | $60^{\circ}$ | Creative Commons License | college_math.precalculus | Convert the angle from radian measure into degree measure: $\frac{\pi}{3}$ | $\boxed{60^{\circ}}$ |
college_math.PRECALCULUS | exercise.3.4.52 | $f(x)=a(x-6)(x-i)(x+i)(x-(1-3 i))(x-(1+3 i))$ where $a$ is any real number, $a<0$ | Creative Commons License | college_math.precalculus | Create a polynomial $f$ that is degree 5 and has the following characteristics:
- $x=6$, $x=i$, and $x=1-3i$ are zeros of $f$
- As $x \rightarrow -\infty$, $f(x) \rightarrow \infty$ | $\boxed{f(x)=a(x-6)(x-i)(x+i)(x-(1-3 i))(x-(1+3 i))$ where $a$ is any real number, $a<0}$ |
college_math.PRECALCULUS | exercise.11.4.23 | $(0,-9)$ | Creative Commons License | college_math.precalculus | Convert the point from polar coordinates into rectangular coordinates: $\left(9, \frac{7 \pi}{2}\right)$ | $\boxed{(0,-9)}$ |
college_math.PRECALCULUS | exercise.10.2.39 | $\cos (\theta)=-1.001$ never happens | Creative Commons License | college_math.precalculus | Find all of the angles which satisfy the given equation: $\cos (\theta)=-1.001$ | $\boxed{\cos (\theta)=-1.001$ never happens}$ |
college_math.PRECALCULUS | exercise.6.3.43 | $(-\infty, 1]$ | Creative Commons License | college_math.precalculus | Use your calculator to help you solve the inequality: $e^{-x}-x e^{-x} \geq 0$ | $\boxed{(-\infty, 1]}$ |
college_math.PRECALCULUS | exercise.5.3.20 | $x=-\frac{1}{3}, \frac{2}{3}$ | Creative Commons License | college_math.precalculus | Solve the equation or inequality: $3 x+\sqrt{6-9 x}=2$ | $\boxed{x=-\frac{1}{3}, \frac{2}{3}}$ |
college_math.PRECALCULUS | exercise.10.7.79 | $\left[0, \frac{\pi}{4}\right] \cup\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right) \cup\left[\frac{7 \pi}{4}, 2 \pi\right]$ | Creative Commons License | college_math.precalculus | Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \leq x \leq 2 \pi$: $\sec (x) \leq \sqrt{2}$ | $\boxed{\left[0, \frac{\pi}{4}\right] \cup\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right) \cup\left[\frac{7 \pi}{4}, 2 \pi\right]}$ |
college_math.PRECALCULUS | exercise.3.3.41 | $x= \pm \sqrt{3}$ | Creative Commons License | college_math.precalculus | Find the real solutions of the polynomial equation $x^{3}+x^{2}=\frac{11 x+10}{3}$. | $\boxed{x= \pm \sqrt{3}}$ |
college_math.PRECALCULUS | exercise.6.2.7 | $3 \log _{\sqrt{2}}(x)+4$ | Creative Commons License | college_math.precalculus | Expand the given logarithm and simplify: $\log _{\sqrt{2}}\left(4 x^{3}\right)$ | $\boxed{3 \log _{\sqrt{2}}(x)+4}$ |
college_math.PRECALCULUS | exercise.2.3.32 | $x= \pm(y-2)$ | Creative Commons License | college_math.precalculus | Solve the quadratic equation $y^{2}-4 y=x^{2}-4$ for $x$. | $\boxed{x= \pm(y-2)}$ |
college_math.PRECALCULUS | exercise.10.2.50 | $\cos (-2.01) \approx-0.425$ | Creative Commons License | college_math.precalculus | Approximate the given value to three decimal places: $\cos (-2.01)$ | $\boxed{\cos (-2.01) \approx-0.425}$ |
college_math.PRECALCULUS | exercise.10.7.32 | $x=\frac{\pi}{6}, \frac{7 \pi}{6}, \frac{5 \pi}{6}, \frac{11 \pi}{6}$ | Creative Commons License | college_math.precalculus | Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\cos (x) \csc (x) \cot (x)=6-\cot ^{2}(x)$ | $\boxed{x=\frac{\pi}{6}, \frac{7 \pi}{6}, \frac{5 \pi}{6}, \frac{11 \pi}{6}}$ |
college_math.PRECALCULUS | exercise.6.4.19 | $x=6$ | Creative Commons License | college_math.precalculus | Solve the equation analytically: $2 \log _{7}(x)=\log _{7}(2)+\log _{7}(x+12)$ | $\boxed{x=6}$ |
college_math.PRECALCULUS | exercise.10.2.63 | The side adjacent to $\theta$ has length $10 \cos \left(5^{\circ}\right) \approx 9.962$. | Creative Commons License | college_math.precalculus | If $\theta=5^{\circ}$ and the hypotenuse has length 10 , how long is the side adjacent to $\theta$ ? | $\boxed{The side adjacent to $\theta$ has length $10 \cos \left(5^{\circ}\right) \approx 9.962$.}$ |
college_math.PRECALCULUS | exercise.6.3.17 | $t=\frac{\ln \left(\frac{1}{18}\right)}{-0.1}=10 \ln (18)$ | Creative Commons License | college_math.precalculus | Solve the equation analytically: $70+90 e^{-0.1 t}=75$ | $\boxed{t=\frac{\ln \left(\frac{1}{18}\right)}{-0.1}=10 \ln (18)}$ |
college_math.PRECALCULUS | exercise.8.7.13 | $(-4,-56),(1,9),(2,16)$ | Creative Commons License | college_math.precalculus | Solve the system of nonlinear equations: $\left\{\begin{array}{rr}y & =x^{3}+8 \\ y & =10 x-x^{2}\end{array}\right.$ | $\boxed{(-4,-56),(1,9),(2,16)}$ |
college_math.PRECALCULUS | exercise.6.1.17 | $\log _{6}(216)=3$ | Creative Commons License | college_math.precalculus | Evaluate the expression: $\log _{6}(216)$ | $\boxed{\log _{6}(216)=3}$ |
college_math.PRECALCULUS | exercise.3.3.35 | $x=0, \frac{5 \pm \sqrt{61}}{18}$ | Creative Commons License | college_math.precalculus | Find the real solutions of the polynomial equation $9 x^{3}=5 x^{2}+x$. | $\boxed{x=0, \frac{5 \pm \sqrt{61}}{18}}$ |
college_math.PRECALCULUS | exercise.6.2.14 | $-2+\frac{2}{3} \log _{\frac{1}{2}}(x)-\log _{\frac{1}{2}}(y)-\frac{1}{2} \log _{\frac{1}{2}}(z)$ | Creative Commons License | college_math.precalculus | Expand the given logarithm and simplify: $\log _{\frac{1}{2}}\left(\frac{4 \sqrt[3]{x^{2}}}{y \sqrt{z}}\right)$ | $\boxed{-2+\frac{2}{3} \log _{\frac{1}{2}}(x)-\log _{\frac{1}{2}}(y)-\frac{1}{2} \log _{\frac{1}{2}}(z)}$ |
college_math.PRECALCULUS | exercise.3.3.44 | $\left\{-\frac{1}{2}\right\} \cup[1, \infty)$ | Creative Commons License | college_math.precalculus | Find the real solutions of the polynomial equation $2 x^{5}+3 x^{4}=18 x+27$. | $\boxed{\left\{-\frac{1}{2}\right\} \cup[1, \infty)}$ |
college_math.PRECALCULUS | exercise.1.1.22 | $d=5, M=\left(-1, \frac{7}{2}\right)$ | Creative Commons License | college_math.precalculus | Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $(1,2),(-3,5)$ | $\boxed{d=5, M=\left(-1, \frac{7}{2}\right)}$ |
college_math.PRECALCULUS | exercise.9.1.1 | $0,1,3,7$ | Creative Commons License | college_math.precalculus | Write out the first four terms of the given sequence: $a_{n}=2^{n}-1, n \geq 0$ | $\boxed{0,1,3,7}$ |
college_math.PRECALCULUS | exercise.9.1.10 | $-2,-\frac{1}{3},-\frac{1}{36},-\frac{1}{720}$ | Creative Commons License | college_math.precalculus | Write out the first four terms of the given sequence: $c_{0}=-2, c_{j}=\frac{c_{j-1}}{(j+1)(j+2)}, j \geq 1$ | $\boxed{-2,-\frac{1}{3},-\frac{1}{36},-\frac{1}{720}}$ |
college_math.PRECALCULUS | exercise.11.4.52 | $\left(\frac{1}{3}, \pi+\arctan (2)\right)$ | Creative Commons License | college_math.precalculus | Convert the equation from rectangular coordinates into polar coordinates: $x^{2}+y^{2}=x$ | $\boxed{\left(\frac{1}{3}, \pi+\arctan (2)\right)}$ |
college_math.PRECALCULUS | exercise.9.2.29 | $\frac{7}{9}$ | Creative Commons License | college_math.precalculus | Express the repeating decimal as a fraction of integers: $0 . \overline{7}$ | $\boxed{\frac{7}{9}}$ |
college_math.PRECALCULUS | exercise.10.7.66 | $x=-1,0$ | Creative Commons License | college_math.precalculus | Solve the equation: $9 \arccos ^{2}(x)-\pi^{2}=0$ | $\boxed{x=-1,0}$ |
college_math.PRECALCULUS | exercise.6.3.21 | $t=\frac{\ln \left(\frac{1}{29}\right)}{-0.8}=\frac{5}{4} \ln (29)$ | Creative Commons License | college_math.precalculus | Solve the equation analytically: $\frac{150}{1+29 e^{-0.8 t}}=75$ | $\boxed{t=\frac{\ln \left(\frac{1}{29}\right)}{-0.8}=\frac{5}{4} \ln (29)}$ |
college_math.PRECALCULUS | exercise.10.7.86 | $\left[-\frac{3 \pi}{4}, \frac{\pi}{4}\right]$ | Creative Commons License | college_math.precalculus | Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-\pi \leq x \leq \pi$: $\cos (x) \geq \sin (x)$ | $\boxed{\left[-\frac{3 \pi}{4}, \frac{\pi}{4}\right]}$ |
college_math.PRECALCULUS | exercise.6.4.27 | $\left[10^{-3}, \infty\right)$ | Creative Commons License | college_math.precalculus | Solve the inequality analytically: $10 \log \left(\frac{x}{10^{-12}}\right) \geq 90$ | $\boxed{\left[10^{-3}, \infty\right)}$ |
college_math.PRECALCULUS | exercise.6.3.4 | $x=-\frac{1}{4}$ | Creative Commons License | college_math.precalculus | Solve the equation analytically: $4^{2 x}=\frac{1}{2}$ | $\boxed{x=-\frac{1}{4}}$ |
college_math.PRECALCULUS | exercise.6.3.34 | $(\ln (53), \infty)$ | Creative Commons License | college_math.precalculus | Solve the inequality analytically: $e^{x}>53$ | $\boxed{(\ln (53), \infty)}$ |
college_math.PRECALCULUS | exercise.10.7.103 | $\bigcup_{k=-\infty}^{\infty}\left(\frac{k \pi}{2}, \frac{(k+1) \pi}{2}\right)$ | Creative Commons License | college_math.precalculus | Express the domain of the function using the extended interval notation: $f(x)=\csc (2 x)$ | $\boxed{\bigcup_{k=-\infty}^{\infty}\left(\frac{k \pi}{2}, \frac{(k+1) \pi}{2}\right)}$ |
college_math.PRECALCULUS | exercise.10.4.80 | $2 \cos (4 \theta) \cos (\theta)$ | Creative Commons License | college_math.precalculus | Write the given sum as a product: $\cos (3 \theta)+\cos (5 \theta)$ | $\boxed{2 \cos (4 \theta) \cos (\theta)}$ |
college_math.PRECALCULUS | exercise.3.3.46 | $\{2\} \cup[4, \infty)$ | Creative Commons License | college_math.precalculus | Solve the polynomial inequality $x^{4}-9 x^{2} \leq 4 x-12$ and state your answer using interval notation. | $\boxed{\{2\} \cup[4, \infty)}$ |
college_math.PRECALCULUS | exercise.11.4.47 | $\left(10, \arctan \left(\frac{4}{3}\right)\right)$ | Creative Commons License | college_math.precalculus | Convert the equation from rectangular coordinates into polar coordinates: $x=3 y+1$ | $\boxed{\left(10, \arctan \left(\frac{4}{3}\right)\right)}$ |
college_math.PRECALCULUS | exercise.8.7.15 | $(3,4)$ | Creative Commons License | college_math.precalculus | Solve the system of nonlinear equations: $\left\{\begin{aligned} x^{2}+y^{2} & =25 \\ 4 x^{2}-9 y & =0 \\ 3 y^{2}-16 x & =0\end{aligned}\right.$ | $\boxed{(3,4)}$ |
college_math.PRECALCULUS | exercise.10.7.78 | $[0,2 \pi]$ | Creative Commons License | college_math.precalculus | Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \leq x \leq 2 \pi$: $\cos (3 x) \leq 1$ | $\boxed{[0,2 \pi]}$ |
college_math.PRECALCULUS | exercise.6.3.9 | $x=\frac{\ln (5)}{2 \ln (3)}$ | Creative Commons License | college_math.precalculus | Solve the equation analytically: $3^{2 x}=5$ | $\boxed{x=\frac{\ln (5)}{2 \ln (3)}}$ |
college_math.PRECALCULUS | exercise.1.1.32 | $(-1+\sqrt{3}, 0),(-1-\sqrt{3}, 0)$ | Creative Commons License | college_math.precalculus | Find all of the points on the $x$-axis which are 2 units from the point $(-1,1)$. | $\boxed{(-1+\sqrt{3}, 0),(-1-\sqrt{3}, 0)}$ |
college_math.PRECALCULUS | exercise.10.7.74 | $\left(0, \frac{\pi}{3}\right] \cup\left[\frac{2 \pi}{3}, \pi\right) \cup\left(\pi, \frac{4 \pi}{3}\right] \cup\left[\frac{5 \pi}{3}, 2 \pi\right)$ | Creative Commons License | college_math.precalculus | Solve the inequality. Express the exact answer in interval notation, restricting your attention to $0 \leq x \leq 2 \pi$: $\sin \left(x+\frac{\pi}{3}\right)>\frac{1}{2}$ | $\boxed{\left(0, \frac{\pi}{3}\right] \cup\left[\frac{2 \pi}{3}, \pi\right) \cup\left(\pi, \frac{4 \pi}{3}\right] \cup\left[\frac{5 \pi}{3}, 2 \pi\right)}$ |
college_math.PRECALCULUS | exercise.3.4.18 | $-3 i$ | Creative Commons License | college_math.precalculus | Simplify the quantity $-\sqrt{(-9)}$ | $\boxed{-3 i}$ |
college_math.PRECALCULUS | exercise.11.4.62 | $\theta=\frac{\pi}{3}$ | Creative Commons License | college_math.precalculus | Convert the equation from polar coordinates into rectangular coordinates: $\theta=\pi$ | $\boxed{\theta=\frac{\pi}{3}}$ |
college_math.PRECALCULUS | exercise.6.3.2 | $x=4$ | Creative Commons License | college_math.precalculus | Solve the equation analytically: $3^{(x-1)}=27$ | $\boxed{x=4}$ |
college_math.PRECALCULUS | exercise.10.7.88 | $[-2 \pi, 2 \pi]$ | Creative Commons License | college_math.precalculus | Solve the inequality. Express the exact answer in interval notation, restricting your attention to $-2 \pi \leq x \leq 2 \pi$: $\cos (x) \leq \frac{5}{3}$ | $\boxed{[-2 \pi, 2 \pi]}$ |
college_math.PRECALCULUS | exercise.6.4.4 | $x=-3,4$ | Creative Commons License | college_math.precalculus | Solve the equation analytically: $\log _{5}\left(18-x^{2}\right)=\log _{5}(6-x)$ | $\boxed{x=-3,4}$ |
college_math.PRECALCULUS | exercise.6.1.34 | $\log _{36}\left(36^{216}\right)=216$ | Creative Commons License | college_math.precalculus | Evaluate the expression: $\log _{36}\left(36^{216}\right)$ | $\boxed{\log _{36}\left(36^{216}\right)=216}$ |
college_math.PRECALCULUS | exercise.8.5.17 | $B^{-1}=\left[\begin{array}{rr}3 & 7 \\ 5 & 12\end{array}\right]$ | Creative Commons License | college_math.precalculus | Find the inverse of the given matrix: $B=\left[\begin{array}{rr}12 & -7 \\ -5 & 3\end{array}\right]$ | $\boxed{B^{-1}=\left[\begin{array}{rr}3 & 7 \\ 5 & 12\end{array}\right]}$ |
college_math.PRECALCULUS | exercise.6.1.56 | No domain | Creative Commons License | college_math.precalculus | Find the domain of the function: $f(x)=\frac{\sqrt{-1-x}}{\log _{\frac{1}{2}}(x)}$ | $\boxed{No domain}$ |
college_math.PRECALCULUS | exercise.2.2.10 | $x=0$ or $x=2$ | Creative Commons License | college_math.precalculus | Solve the equation: $|2 x-1|=x+1$ | $\boxed{x=0$ or $x=2}$ |
college_math.PRECALCULUS | exercise.6.3.29 | $x=\ln (3)$ | Creative Commons License | college_math.precalculus | Solve the equation analytically: $e^{2 x}=e^{x}+6$ | $\boxed{x=\ln (3)}$ |
college_math.PRECALCULUS | exercise.10.7.49 | $x=\frac{\pi}{12}, \frac{17 \pi}{12}$ | Creative Commons License | college_math.precalculus | Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\sqrt{2} \cos (x)-\sqrt{2} \sin (x)=1$ | $\boxed{x=\frac{\pi}{12}, \frac{17 \pi}{12}}$ |
college_math.PRECALCULUS | exercise.9.4.17 | -1 | Creative Commons License | college_math.precalculus | Simplify the power of a complex number: $\left(\frac{\sqrt{2}}{2}-\frac{\sqrt{2}}{2} i\right)^{4}$ | $\boxed{-1}$ |
college_math.PRECALCULUS | exercise.6.4.23 | $x=10^{-3}, 10^{5}$ | Creative Commons License | college_math.precalculus | Solve the equation analytically: $(\log (x))^{2}=2 \log (x)+15$ | $\boxed{x=10^{-3}, 10^{5}}$ |
college_math.PRECALCULUS | exercise.4.3.11 | $(-\infty,-3) \cup(-3,2) \cup(4, \infty)$ | Creative Commons License | college_math.precalculus | Solve the rational inequality and express your answer using interval notation: $\frac{x^{2}-x-12}{x^{2}+x-6}>0$ | $\boxed{(-\infty,-3) \cup(-3,2) \cup(4, \infty)}$ |
college_math.PRECALCULUS | exercise.8.2.22 | $\left(\frac{19}{13} t+\frac{51}{13},-\frac{11}{13} t+\frac{4}{13}, t\right)$ for all real numbers $t$ | Creative Commons License | college_math.precalculus | Solve the following system of linear equations: $\left\{\begin{aligned} x-3 y-4 z & =3 \\ 3 x+4 y-z & =13 \\ 2 x-19 y-19 z & =2\end{aligned}\right.$ | $\boxed{\left(\frac{19}{13} t+\frac{51}{13},-\frac{11}{13} t+\frac{4}{13}, t\right)$ for all real numbers $t}$ |
college_math.PRECALCULUS | exercise.6.3.1 | $x=\frac{3}{4}$ | Creative Commons License | college_math.precalculus | Solve the equation analytically: $2^{4 x}=8$ | $\boxed{x=\frac{3}{4}}$ |
college_math.PRECALCULUS | exercise.6.4.28 | $\left[10^{2.6}, 10^{4.1}\right]$ | Creative Commons License | college_math.precalculus | Solve the inequality analytically: $5.6 \leq \log \left(\frac{x}{10^{-3}}\right) \leq 7.1$ | $\boxed{\left[10^{2.6}, 10^{4.1}\right]}$ |
college_math.PRECALCULUS | exercise.6.3.42 | $x=0$ | Creative Commons License | college_math.precalculus | Use your calculator to help you solve the equation: $e^{\sqrt{x}}=x+1$ | $\boxed{x=0}$ |
college_math.PRECALCULUS | exercise.6.4.29 | $\left(10^{-5.4}, 10^{-2.3}\right)$ | Creative Commons License | college_math.precalculus | Solve the inequality analytically: $2.3<-\log (x)<5.4$ | $\boxed{\left(10^{-5.4}, 10^{-2.3}\right)}$ |
college_math.PRECALCULUS | exercise.8.4.2 | $B^{-1}=\left[\begin{array}{rr}3 & 7 \\ 5 & 12\end{array}\right]$ | Creative Commons License | college_math.precalculus | Find the inverse of the matrix or state that the matrix is not invertible: $B=\left[\begin{array}{rr}12 & -7 \\ -5 & 3\end{array}\right]$ | $\boxed{B^{-1}=\left[\begin{array}{rr}3 & 7 \\ 5 & 12\end{array}\right]}$ |
college_math.PRECALCULUS | exercise.6.2.29 | $\log _{2}\left(\frac{x}{x-1}\right)$ | Creative Commons License | college_math.precalculus | Use the properties of logarithms to write the expression as a single logarithm: $\log _{2}(x)+\log _{\frac{1}{2}}(x-1)$ | $\boxed{\log _{2}\left(\frac{x}{x-1}\right)}$ |
college_math.PRECALCULUS | exercise.6.3.35 | $\left[\frac{\ln (3)}{12 \ln (1.005)}, \infty\right)$ | Creative Commons License | college_math.precalculus | Solve the inequality analytically: $1000(1.005)^{12 t} \geq 3000$ | $\boxed{\left[\frac{\ln (3)}{12 \ln (1.005)}, \infty\right)}$ |
college_math.PRECALCULUS | exercise.6.2.25 | $\log _{7}\left(\frac{x(x-3)}{49}\right)$ | Creative Commons License | college_math.precalculus | Use the properties of logarithms to write the expression as a single logarithm: $\log _{7}(x)+\log _{7}(x-3)-2$ | $\boxed{\log _{7}\left(\frac{x(x-3)}{49}\right)}$ |
college_math.PRECALCULUS | exercise.10.7.21 | $x=\frac{\pi}{6}, \frac{\pi}{2}, \frac{5 \pi}{6}, \frac{3 \pi}{2}$ | Creative Commons License | college_math.precalculus | Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\sin (2 x)=\cos (x)$ | $\boxed{x=\frac{\pi}{6}, \frac{\pi}{2}, \frac{5 \pi}{6}, \frac{3 \pi}{2}}$ |
college_math.PRECALCULUS | exercise.10.2.59 | The hypotenuse has length $\frac{4}{\cos \left(12^{\circ}\right)} \approx 4.089$. | Creative Commons License | college_math.precalculus | If $\theta=12^{\circ}$ and the side adjacent to $\theta$ has length 4 , how long is the hypotenuse? | $\boxed{The hypotenuse has length $\frac{4}{\cos \left(12^{\circ}\right)} \approx 4.089$.}$ |
college_math.PRECALCULUS | exercise.10.2.19 | $\cos \left(\frac{10 \pi}{3}\right)=-\frac{1}{2}, \sin \left(\frac{10 \pi}{3}\right)=-\frac{\sqrt{3}}{2}$ | Creative Commons License | college_math.precalculus | Find the exact value of the cosine and sine of the given angle: $\theta=\frac{10 \pi}{3}$ | $\boxed{\cos \left(\frac{10 \pi}{3}\right)=-\frac{1}{2}, \sin \left(\frac{10 \pi}{3}\right)=-\frac{\sqrt{3}}{2}}$ |
college_math.PRECALCULUS | exercise.1.1.34 | (-3, -4), 5 miles, $(4,-4)$ | Creative Commons License | college_math.precalculus | Let's assume for a moment that we are standing at the origin and the positive $y$-axis points due North while the positive $x$-axis points due East. Our Sasquatch-o-meter tells us that Sasquatch is 3 miles West and 4 miles South of our current position. What are the coordinates of his position? How far away is he from us? If he runs 7 miles due East what would his new position be? | $\boxed{(-3, -4), 5 miles, $(4,-4)}$ |
college_math.PRECALCULUS | exercise.8.2.8 | $(-3,20,19)$ | Creative Commons License | college_math.precalculus | Solve the following system of linear equations: $\left\{\begin{aligned} x+y+z & =3 \\ 2 x-y+z & =0 \\ -3 x+5 y+7 z & =7\end{aligned}\right.$ | $\boxed{(-3,20,19)}$ |
college_math.PRECALCULUS | exercise.1.1.26 | $d=\sqrt{74}, M=\left(\frac{13}{10},-\frac{13}{10}\right)$ | Creative Commons License | college_math.precalculus | Find the distance $d$ between the points and the midpoint $M$ of the line segment which connects them: $\left(\frac{24}{5}, \frac{6}{5}\right),\left(-\frac{11}{5},-\frac{19}{5}\right)$. | $\boxed{d=\sqrt{74}, M=\left(\frac{13}{10},-\frac{13}{10}\right)}$ |
college_math.PRECALCULUS | exercise.10.7.8 | No solution | Creative Commons License | college_math.precalculus | Find all of the exact solutions of the equation and then list those solutions which are in the interval $[0,2 \pi)$: $\cos (9 x)=9$ | $\boxed{No solution}$ |
college_math.PRECALCULUS | exercise.6.4.33 | $\approx(-\infty,-12.1414) \cup(12.1414, \infty)$ | Creative Commons License | college_math.precalculus | Solve the equation or inequality using your calculator: $\ln \left(x^{2}+1\right) \geq 5$ | $\boxed{\approx(-\infty,-12.1414) \cup(12.1414, \infty)}$ |
college_math.PRECALCULUS | exercise.10.2.7 | $\cos (\pi)=-1, \sin (\pi)=0$ | Creative Commons License | college_math.precalculus | Find the exact value of the cosine and sine of the given angle: $\theta=\pi$ | $\boxed{\cos (\pi)=-1, \sin (\pi)=0}$ |
college_math.PRECALCULUS | exercise.10.2.53 | $\sin \left(\pi^{\circ}\right) \approx 0.055$ | Creative Commons License | college_math.precalculus | Approximate the given value to three decimal places: $\sin \left(\pi^{\circ}\right)$ | $\boxed{\sin \left(\pi^{\circ}\right) \approx 0.055}$ |
college_math.PRECALCULUS | exercise.8.4.11 | $\left[\begin{array}{rr}12 & -7 \\ -5 & 3\end{array}\right]\left[\begin{array}{r}-7 \\ 5\end{array}\right]=\left[\begin{array}{r}-119 \\ 50\end{array}\right]$ So $x=-119$ and $y=50$. | Creative Commons License | college_math.precalculus | Use one matrix inverse to solve the following system of linear equations:
$\left\{\begin{aligned} 3 x+7 y & =-7 \\ 5 x+12 y & =5\end{aligned}\right.$ | $\boxed{\left[\begin{array}{rr}12 & -7 \\ -5 & 3\end{array}\right]\left[\begin{array}{r}-7 \\ 5\end{array}\right]=\left[\begin{array}{r}-119 \\ 50\end{array}\right]$ So $x=-119$ and $y=50$.}$ |
college_math.PRECALCULUS | exercise.8.4.5 | $E^{-1}=\left[\begin{array}{rrr}-1 & 8 & 4 \\ 1 & -3 & -1 \\ 1 & -6 & -3\end{array}\right]$ | Creative Commons License | college_math.precalculus | Find the inverse of the matrix or state that the matrix is not invertible: $E=\left[\begin{array}{rrr}3 & 0 & 4 \\ 2 & -1 & 3 \\ -3 & 2 & -5\end{array}\right]$ | $\boxed{E^{-1}=\left[\begin{array}{rrr}-1 & 8 & 4 \\ 1 & -3 & -1 \\ 1 & -6 & -3\end{array}\right]}$ |
college_math.PRECALCULUS | exercise.6.2.24 | $\log \left(\frac{1000}{x}\right)$ | Creative Commons License | college_math.precalculus | Use the properties of logarithms to write the expression as a single logarithm: $3-\log (x)$ | $\boxed{\log \left(\frac{1000}{x}\right)}$ |
college_math.PRECALCULUS | exercise.8.2.23 | Inconsistent | Creative Commons License | college_math.precalculus | Solve the following system of linear equations: $\left\{\begin{aligned} x+y+z & =4 \\ 2 x-4 y-z & =-1 \\ x-y & =2\end{aligned}\right.$ | $\boxed{Inconsistent}$ |
college_math.PRECALCULUS | exercise.10.7.20 | $x=0, \frac{\pi}{3}, \pi, \frac{5 \pi}{3}$ | Creative Commons License | college_math.precalculus | Solve the equation, giving the exact solutions which lie in $[0,2 \pi)$: $\sin (2 x)=\sin (x)$ | $\boxed{x=0, \frac{\pi}{3}, \pi, \frac{5 \pi}{3}}$ |
college_math.PRECALCULUS | exercise.3.4.25 | $i^{117}=\left(i^{4}\right)^{29} \cdot i=1 \cdot i=i$ | Creative Commons License | college_math.precalculus | Simplify the given power of $i$: $i^{117}$ | $\boxed{i^{117}=\left(i^{4}\right)^{29} \cdot i=1 \cdot i=i}$ |
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