Solve for x cube root of 8^x=65536

$\sqrt[3]{8^{x}} = 65536$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{8^{x}}$ as $8^{\frac{x}{3}}$ .

$8^{\frac{x}{3}} = 65536$

Step 2

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

$2^{3 \frac{x}{3}} = 2^{16}$

Step 3

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

$3 \frac{x}{3} = 16$

Step 4

Solve for $x$ .

Combine $3$ and $\frac{x}{3}$ .

$\frac{3 x}{3} = 16$

Divide each term in $\frac{3 x}{3} = 16$ by $3$ and simplify.

Divide each term in $\frac{3 x}{3} = 16$ by $3$ .

$\frac{\frac{3 x}{3}}{3} = \frac{16}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{\frac{3 x}{3}}{3} = \frac{16}{3}$

Divide $x$ by $1$ .

$\frac{x}{3} = \frac{16}{3}$

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = 16$