Solve for x 25 sixth root of 25=5^x

$25 \sqrt[6]{25} = 5^{x}$

Step 1

Rewrite the equation as $5^{x} = 25 \sqrt[6]{25}$ .

$5^{x} = 25 \sqrt[6]{25}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{25}$ as $25^{\frac{1}{6}}$ .

$5^{x} = 25 \cdot 25^{\frac{1}{6}}$

Step 3

Multiply $25$ by $25^{\frac{1}{6}}$ by adding the exponents.

Multiply $25$ by $25^{\frac{1}{6}}$ .

Raise $25$ to the power of $1$ .

$5^{x} = 25^{1} \cdot 25^{\frac{1}{6}}$

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

$5^{x} = 25^{1 + \frac{1}{6}}$

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

$5^{x} = 25^{\frac{6}{6} + \frac{1}{6}}$

Combine the numerators over the common denominator.

$5^{x} = 25^{\frac{6 + 1}{6}}$

Add $6$ and $1$ .

$5^{x} = 25^{\frac{7}{6}}$

Step 4

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

$5^{x} = 5^{2 (\frac{7}{6})}$

Step 5

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

$x = 2 (\frac{7}{6})$

Step 6

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$x = 2 \frac{7}{2 (3)}$

Cancel the common factor.

$x = 2 \frac{7}{2 \cdot 3}$

Rewrite the expression.

$x = \frac{7}{3}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{3}$

Decimal Form:

$x = 2.\overline{3}$

Mixed Number Form:

$x = 2 \frac{1}{3}$