Precalculus Examples

$5^{p + 2} = 5^{- 3 p}$

Step 1

Create equivalent expressions in the equation that all have equal bases.

$5^{p + 2} = 5^{- 3 p}$

Step 2

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$p + 2 = - 3 p$

Step 3

Solve for $p$ .

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Step 3.1

Move all terms containing $p$ to the left side of the equation.

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Step 3.1.1

Add $3 p$ to both sides of the equation.

$p + 2 + 3 p = 0$

Step 3.1.2

Add $p$ and $3 p$ .

$4 p + 2 = 0$

Step 3.2

Subtract $2$ from both sides of the equation.

$4 p = - 2$

Step 3.3

Divide each term in $4 p = - 2$ by $4$ and simplify.

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Step 3.3.1

Divide each term in $4 p = - 2$ by $4$ .

$\frac{4 p}{4} = \frac{- 2}{4}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $4$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{4 p}{4} = \frac{- 2}{4}$

Step 3.3.2.1.2

Divide $p$ by $1$ .

$p = \frac{- 2}{4}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Cancel the common factor of $- 2$ and $4$ .

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Step 3.3.3.1.1

Factor $2$ out of $- 2$ .

$p = \frac{2 (- 1)}{4}$

Step 3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.1.2.1

Factor $2$ out of $4$ .

$p = \frac{2 \cdot - 1}{2 \cdot 2}$

Step 3.3.3.1.2.2

Cancel the common factor.

$p = \frac{2 \cdot - 1}{2 \cdot 2}$

Step 3.3.3.1.2.3

Rewrite the expression.

$p = \frac{- 1}{2}$

Step 3.3.3.2

Move the negative in front of the fraction.

$p = - \frac{1}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$p = - \frac{1}{2}$

Decimal Form:

$p = - 0.5$