Precalculus Examples

$25^{- a} \cdot 625 = 625^{3 a}$

Step 1

Rewrite $25$ as $5^{2}$ .

${(5^{2})}^{- a} \cdot 625 = 625^{3 a}$

Step 2

Multiply the exponents in ${(5^{2})}^{- a}$ .

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Step 2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$5^{2 (- a)} \cdot 625 = 625^{3 a}$

Step 2.2

Multiply $- 1$ by $2$ .

$5^{- 2 a} \cdot 625 = 625^{3 a}$

Step 3

Rewrite $625$ as $5^{4}$ .

$5^{- 2 a} \cdot 5^{4} = 625^{3 a}$

Step 4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5^{- 2 a + 4} = 625^{3 a}$

Step 5

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

$5^{- 2 a + 4} = 5^{4 (3 a)}$

Step 6

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

$- 2 a + 4 = 4 (3 a)$

Step 7

Solve for $a$ .

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Step 7.1

Multiply $3$ by $4$ .

$- 2 a + 4 = 12 a$

Step 7.2

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

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Step 7.2.1

Subtract $12 a$ from both sides of the equation.

$- 2 a + 4 - 12 a = 0$

Step 7.2.2

Subtract $12 a$ from $- 2 a$ .

$- 14 a + 4 = 0$

Step 7.3

Subtract $4$ from both sides of the equation.

$- 14 a = - 4$

Step 7.4

Divide each term in $- 14 a = - 4$ by $- 14$ and simplify.

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Step 7.4.1

Divide each term in $- 14 a = - 4$ by $- 14$ .

$\frac{- 14 a}{- 14} = \frac{- 4}{- 14}$

Step 7.4.2

Simplify the left side.

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Step 7.4.2.1

Cancel the common factor of $- 14$ .

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Step 7.4.2.1.1

Cancel the common factor.

$\frac{- 14 a}{- 14} = \frac{- 4}{- 14}$

Step 7.4.2.1.2

Divide $a$ by $1$ .

$a = \frac{- 4}{- 14}$

Step 7.4.3

Simplify the right side.

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Step 7.4.3.1

Cancel the common factor of $- 4$ and $- 14$ .

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Step 7.4.3.1.1

Factor $- 2$ out of $- 4$ .

$a = \frac{- 2 (2)}{- 14}$

Step 7.4.3.1.2

Cancel the common factors.

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Step 7.4.3.1.2.1

Factor $- 2$ out of $- 14$ .

$a = \frac{- 2 \cdot 2}{- 2 \cdot 7}$

Step 7.4.3.1.2.2

Cancel the common factor.

$a = \frac{- 2 \cdot 2}{- 2 \cdot 7}$

Step 7.4.3.1.2.3

Rewrite the expression.

$a = \frac{2}{7}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$a = \frac{2}{7}$

Decimal Form:

$a = 0.\overline{285714}$