Solve for x 3^x=3 square root of 3

$3^{x} = 3 \sqrt{3}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 2

Multiply $3$ by $3^{\frac{1}{2}}$ by adding the exponents.

Multiply $3$ by $3^{\frac{1}{2}}$ .

Raise $3$ to the power of $1$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $2$ and $1$ .

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

Step 3

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 1.5$

Mixed Number Form:

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