Solve for x 2 fourth root of 8=2^x

$2 \sqrt[4]{8} = 2^{x}$

Step 1

Rewrite the equation as $2^{x} = 2 \sqrt[4]{8}$ .

$2^{x} = 2 \sqrt[4]{8}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{8}$ as $8^{\frac{1}{4}}$ .

$2^{x} = 2 \cdot 8^{\frac{1}{4}}$

Step 3

Rewrite $8$ as $2^{3}$ .

$2^{x} = 2 \cdot {(2^{3})}^{\frac{1}{4}}$

Step 4

Multiply the exponents in ${(2^{3})}^{\frac{1}{4}}$ .

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

$2^{x} = 2 \cdot 2^{3 (\frac{1}{4})}$

Combine $3$ and $\frac{1}{4}$ .

$2^{x} = 2 \cdot 2^{\frac{3}{4}}$

Step 5

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

$2^{x} = 2^{1 + \frac{3}{4}}$

Step 6

Write $1$ as a fraction with a common denominator.

$2^{x} = 2^{\frac{4}{4} + \frac{3}{4}}$

Step 7

Combine the numerators over the common denominator.

$2^{x} = 2^{\frac{4 + 3}{4}}$

Step 8

Add $4$ and $3$ .

$2^{x} = 2^{\frac{7}{4}}$

Step 9

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

$x = \frac{7}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{4}$

Decimal Form:

$x = 1.75$

Mixed Number Form:

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