Precalculus Examples

${(\frac{1}{8})}^{- 3 a} = 512^{3 a}$

Step 1

Apply the product rule to $\frac{1}{8}$ .

$\frac{1^{- 3 a}}{8^{- 3 a}} = 512^{3 a}$

Step 2

One to any power is one.

$\frac{1}{8^{- 3 a}} = 512^{3 a}$

Step 3

Move $8^{- 3 a}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$8^{3 a} = 512^{3 a}$

Step 4

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

$8^{3 a} = 8^{3 (3 a)}$

Step 5

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

$3 a = 3 (3 a)$

Step 6

Solve for $a$ .

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Step 6.1

Multiply $3$ by $3$ .

$3 a = 9 a$

Step 6.2

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

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Step 6.2.1

Subtract $9 a$ from both sides of the equation.

$3 a - 9 a = 0$

Step 6.2.2

Subtract $9 a$ from $3 a$ .

$- 6 a = 0$

Step 6.3

Divide each term in $- 6 a = 0$ by $- 6$ and simplify.

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Step 6.3.1

Divide each term in $- 6 a = 0$ by $- 6$ .

$\frac{- 6 a}{- 6} = \frac{0}{- 6}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $- 6$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{- 6 a}{- 6} = \frac{0}{- 6}$

Step 6.3.2.1.2

Divide $a$ by $1$ .

$a = \frac{0}{- 6}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Divide $0$ by $- 6$ .

$a = 0$