Solve for a 1/64=16^(2a)

$\frac{1}{64} = 16^{2 a}$

Step 1

Rewrite the equation as $16^{2 a} = \frac{1}{64}$ .

$16^{2 a} = \frac{1}{64}$

Step 2

Raise $64$ to the power of $1$ .

$16^{2 a} = \frac{1}{64^{1}}$

Step 3

Move $64^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$16^{2 a} = 64^{- 1}$

Step 4

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

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

Step 5

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

$2 (2 a) = - 3$

Step 6

Solve for $a$ .

Multiply $2$ by $2$ .

$4 \cdot a = - 3$

Divide each term in $(4 \cdot a) = - 3$ by $4$ and simplify.

Divide each term in $(4 \cdot a) = - 3$ by $4$ .

$\frac{(4 \cdot a)}{4} = \frac{- 3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 \cdot a}{4} = \frac{- 3}{4}$

Divide $a$ by $1$ .

$a = \frac{- 3}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$a = - \frac{3}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = - \frac{3}{4}$

Decimal Form:

$a = - 0.75$