Solve for x ((3^x)^(1/2))/(3^(1/2))=3

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

Step 1

Multiply both sides of the equation by $3^{\frac{1}{2}}$ .

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

Step 2

Simplify both sides of the equation.

Simplify the left side.

Simplify $3^{\frac{1}{2}} \frac{{(3^{x})}^{\frac{1}{2}}}{3^{\frac{1}{2}}}$ .

Cancel the common factor of $3^{\frac{1}{2}}$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Combine $x$ and $\frac{1}{2}$ .

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

Simplify the right side.

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $2$ .

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

Step 3

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

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

Step 4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$x = 3$