Solve for x 8^x=2^5*4^4

$8^{x} = 2^{5} \cdot 4^{4}$

Step 1

Rewrite $4$ as $2^{2}$ .

$8^{x} = 2^{5} \cdot {(2^{2})}^{4}$

Step 2

Multiply the exponents in ${(2^{2})}^{4}$ .

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

$8^{x} = 2^{5} \cdot 2^{2 \cdot 4}$

Multiply $2$ by $4$ .

$8^{x} = 2^{5} \cdot 2^{8}$

Step 3

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

$8^{x} = 2^{5 + 8}$

Step 4

Add $5$ and $8$ .

$8^{x} = 2^{13}$

Step 5

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

$2^{3 x} = 2^{13}$

Step 6

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

$3 x = 13$

Step 7

Divide each term in $3 x = 13$ by $3$ and simplify.

Divide each term in $3 x = 13$ by $3$ .

$\frac{3 x}{3} = \frac{13}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{13}{3}$

Divide $x$ by $1$ .

$x = \frac{13}{3}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{13}{3}$

Decimal Form:

$x = 4.\overline{3}$

Mixed Number Form:

$x = 4 \frac{1}{3}$