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Compare linear, exponential, and quadratic growth

Solve the exponential equation for

x

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\begin{array}{l} \left(\frac{1}{81}\right)^{2 x+9}=27^{6 x-7} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} 2^{6 x-8} \cdot 64^{2-x}=2^{9 x+4} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} \left(\frac{1}{25}\right)^{2 x+3}=125^{9 x+8} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} 2^{6 x+7}=\left(\frac{1}{8}\right)^{1-5 x} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} 3^{3 x-2}=9^{4 x-1} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} \left(\frac{1}{4}\right)^{9 x-5}=32^{x+8} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} \frac{5^{4 x+3}}{25^{9-x}}=5^{2 x+5} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} 3^{7 x+4}=\left(\frac{1}{27}\right)^{x-3} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} 8^{5 x+3}=\left(\frac{1}{64}\right)^{-\frac{2}{3}} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} 27^{3 x-7}=9^{6 x-1} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} \frac{49^{4 x-1}}{7^{x-6}}=7^{8 x+3} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} 64^{x-6} \cdot 4^{9 x-3}=64^{6 x+4} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} 2^{4 x-5} \cdot 32^{x+1}=2^{6 x-7} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} 27^{5 x-4}=81^{7 x-2} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} 4^{x-10}=\left(\frac{1}{64}\right)^{5 x+2} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} \left(\frac{1}{64}\right)^{-8 x+3}=\left(\frac{1}{16}\right)^{7 x-2} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} 2^{3 x+5}=64^{x-7} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} 2^{9 x+2}=16^{5 x-2} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} \frac{2^{8-x}}{32^{x+12}}=2^{3 x-7} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} 3^{4 x+5}=81^{-\frac{3}{4}} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} \frac{2^{3 x-4}}{4^{x-5}}=2^{8 x-1} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} \frac{81^{x+5}}{3^{7 x+2}}=3^{x-2} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} \left(\frac{625}{256}\right)^{5 x-4}=1 \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} 2^{5 x+8}=\left(\frac{1}{32}\right)^{3} \\ x=\square \end{array}

Solve the exponential equation for

x

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\begin{array}{l} 4^{9-6 x}=\left(\frac{1}{64}\right)^{\frac{10}{3}} \\ x=\square \end{array}

The geometric sequence

a_{i}

is defined by the formula:

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\begin{array}{l} a_{1}=-\frac{1}{3} \\ a_{i}=a_{i-1} \cdot(-3) \end{array}

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Find the sum of the first

75

terms in the sequence.

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1

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-1.01 \cdot 10^{35}

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-6.76 \cdot 10^{34}

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-5.06 \cdot 10^{34}

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1.69 \cdot 10^{34}

Consider the complex number

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z=6\left(\cos \left(165^{\circ}\right)+i \sin \left(165^{\circ}\right)\right) \text {. }

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Which of the following complex numbers best approximates

z

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1

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-5.8-1.6 i

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-1.6+5.8 i

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-5.8+1.6 i

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-1.6-5.8 i

Consider the complex number

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z=3\left(\cos \left(355^{\circ}\right)+i \sin \left(355^{\circ}\right)\right) \text {. }

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Which of the following complex numbers best approximates

z

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1

\newline

3-0.3 i

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0.3+3 i

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3+0.3 i

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0.3-3 i

Consider the complex number

z=8\left(\cos \left(240^{\circ}\right)+i \sin \left(240^{\circ}\right)\right) \text {. }

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Which of the following complex numbers best approximates

z

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1

\newline

6.9-4 i

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-6.9-4 i

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-4-6.9 i

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4-6.9 i

Consider the complex number

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z=4\left(\cos \left(120^{\circ}\right)+i \sin \left(120^{\circ}\right)\right) \text {. }

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Which of the following complex numbers best approximates

z

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1

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3.5-2 i

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2-3.5 i

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-2+3.5 i

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-3.5+2 i

The equation of an ellipse is given below.

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\frac{(x+13)^{2}}{400}+\frac{(y+6)^{2}}{625}=1

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What are the foci of this ellipse?

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1

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(-13+\sqrt{15},-6)

(-13-\sqrt{15},-6)

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(-13,-21)

(-13,9)

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(-28,-6)

(2,-6)

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(-13,-6+\sqrt{15})

(-13,-6-\sqrt{15})

The equation of an ellipse is given below.

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\frac{(x+7)^{2}}{8}+\frac{(y-10)^{2}}{13}=1

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What are the foci of this ellipse?

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1

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(-7+\sqrt{5}, 10)

(-7-\sqrt{5}, 10)

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(7,-10+\sqrt{105})

(7,-10-\sqrt{105})

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(7+\sqrt{105},-10)

(7-\sqrt{105},-10)

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(-7,10+\sqrt{5})

(-7,10-\sqrt{5})

\begin{array}{l} g(x)=9 x-2 \\ h(x)=3 x^{2}+2 x \end{array}

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h(g(x))

as an expression in terms of

x

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h(g(x))=

\begin{array}{l} g(x)=x^{2}+6 x \\ h(x)=x^{2}-1 \end{array}

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g(h(x))

as an expression in terms of

x

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g(h(x))=

\begin{array}{l}k(x)=6 x+100 \\ k(-5)=\square\end{array}

Consider the equation

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0.75 \cdot 10^{\frac{w}{3}}=30 \text {. }

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Solve the equation for

w

. Express the solution as a logarithm in base

10

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w=

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Approximate the value of

w

. Round your answer to the nearest thousandth.

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w \approx

-7 \cdot 3^{0.25 x}=-10

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Which of the following is the solution of the equation?

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1

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x=\frac{4 \log _{3}(10)}{\log _{3}(7)}

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x=4 \log _{3}\left(\frac{10}{7}\right)

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x=\frac{4 \log _{3}(7)}{\log _{3} 10}

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x=4 \log _{\frac{10}{7}}(3)

Evaluate the logarithm.

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Round your answer to the nearest thousandth.

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\log _{5}\left(\frac{1}{1000}\right) \approx \square

Evaluate the logarithm.

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Round your answer to the nearest thousandth.

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\log _{3}(0.2) \approx

Evaluate the logarithm.

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Round your answer to the nearest thousandth.

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\log _{2}(75) \approx

Evaluate the logarithm.

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Round your answer to the nearest thousandth.

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\log _{4}(0.6) \approx

Evaluate the logarithm.

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Round your answer to the nearest thousandth.

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2 \log _{6}(33) \approx

Evaluate the logarithm.

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Round your answer to the nearest thousandth.

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2 \log _{3}\left(\frac{1}{52}\right) \approx \square

Evaluate the logarithm.

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Round your answer to the nearest thousandth.

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4 \log _{5}(413) \approx

Evaluate the logarithm.

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Round your answer to the nearest thousandth.

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\log _{3}\left(\frac{1}{45}\right) \approx

Evaluate the logarithm.

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Round your answer to the nearest thousandth.

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3 \log _{9}\left(\frac{1}{12}\right) \approx \square

Evaluate the logarithm.

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Round your answer to the nearest thousandth.

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\log _{8}(5) \approx

Evaluate the logarithm.

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Round your answer to the nearest thousandth.

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\log _{8}(500) \approx

Evaluate the logarithm.

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Round your answer to the nearest thousandth.

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\log _{3}(346) \approx

Evaluate the logarithm.

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Round your answer to the nearest thousandth.

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\log _{5}(200) \approx