### About Exponential Functions:

When we work with exponential equations, we first ensure we have the same base on each side. In many cases, we must create an equivalent equation where the base is the same on each side. We can then simplify the exponents, set them equal to each other, and solve. We check the result in the original equation.

Test Objectives

- Demonstrate the ability to rewrite an exponential equation as an equivalent equation where each side has the same base
- Demonstrate the ability to simplify the exponents and create an equation
- Demonstrate the ability to solve the equation and check the solution

#1:

Instructions: Solve each equation.

a) $$5^{v + 2}=25$$

b) $$3^{2x - 3}=3^{-3x}$$

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#2:

Instructions: Solve each equation.

a) $$\left(\frac{1}{81}\right)^{-2r + 1}=\hspace{.25em}9^{2r}$$

b) $$125^{3 - x}=25$$

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#3:

Instructions: Solve each equation.

a) $$32 \cdot 64^{x - 2}=8^{2x}$$

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#4:

Instructions: Solve each equation.

a) $$216^{x + 3}\cdot 36^{-2x + 1}=216$$

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#5:

Instructions: Solve each equation.

a) $$\frac{\left(\frac{1}{9}\right)^{1 - 3x}}{9^{2x}}\hspace{.25em}=\hspace{.25em}27^{3x + 2}$$

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Written Solutions:

#1:

Solutions:

a) $$v=0$$

b) $$x=\frac{3}{5}$$

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#2:

Solutions:

a) $$r=1$$

b) $$x=\frac{7}{3}$$

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#3:

Solutions:

a) $$no\hspace{.25em}solution \hspace{.25em}: \hspace{.25em}∅$$

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#4:

Solutions:

a) $$x=8$$

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#5:

Solutions:

a) $$x=-\frac{8}{7}$$