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

$5^{x} = 25 \sqrt{5}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$5^{x} = 25 \cdot 5^{\frac{1}{2}}$

Step 2

Rewrite $25$ as $5^{2}$ .

$5^{x} = 5^{2} \cdot 5^{\frac{1}{2}}$

Step 3

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

$5^{x} = 5^{2 + \frac{1}{2}}$

Step 4

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$5^{x} = 5^{2 \cdot \frac{2}{2} + \frac{1}{2}}$

Step 5

Combine $2$ and $\frac{2}{2}$ .

$5^{x} = 5^{\frac{2 \cdot 2}{2} + \frac{1}{2}}$

Step 6

Combine the numerators over the common denominator.

$5^{x} = 5^{\frac{2 \cdot 2 + 1}{2}}$

Step 7

Simplify the numerator.

Multiply $2$ by $2$ .

$5^{x} = 5^{\frac{4 + 1}{2}}$

Add $4$ and $1$ .

$5^{x} = 5^{\frac{5}{2}}$

Step 8

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

$x = \frac{5}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{2}$

Decimal Form:

$x = 2.5$

Mixed Number Form:

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