Solve for x 256^(4x)=64^(x+2)

$256^{4 x} = 64^{x + 2}$

Step 1

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

$4^{4 (4 x)} = 4^{3 (x + 2)}$

Step 2

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

$4 (4 x) = 3 (x + 2)$

Step 3

Solve for $x$ .

Multiply $4$ by $4$ .

$16 x = 3 (x + 2)$

Simplify $3 (x + 2)$ .

Apply the distributive property.

$16 x = 3 x + 3 \cdot 2$

Multiply $3$ by $2$ .

$16 x = 3 x + 6$

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

Subtract $3 x$ from both sides of the equation.

$16 x - 3 x = 6$

Subtract $3 x$ from $16 x$ .

$13 x = 6$

Divide each term in $13 x = 6$ by $13$ and simplify.

Divide each term in $13 x = 6$ by $13$ .

$\frac{13 x}{13} = \frac{6}{13}$

Simplify the left side.

Cancel the common factor of $13$ .

Cancel the common factor.

$\frac{13 x}{13} = \frac{6}{13}$

Divide $x$ by $1$ .

$x = \frac{6}{13}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{6}{13}$

Decimal Form:

$x = 0.\overline{461538}$