Precalculus Examples

${(\sqrt{3})}^{x} = 9$

Step 1

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

${(3^{\frac{1}{2}})}^{x} = 9$

Step 2

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

$3^{\frac{1}{2} x} = 9$

Step 3

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

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

Step 4

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

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

Step 5

Solve for $x$ .

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Step 5.1

Multiply both sides of the equation by $2$ .

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

Step 5.2

Simplify both sides of the equation.

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Step 5.2.1

Simplify the left side.

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Step 5.2.1.1

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

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Step 5.2.1.1.1

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

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

Step 5.2.1.1.2

Cancel the common factor of $2$ .

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Step 5.2.1.1.2.1

Cancel the common factor.

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

Step 5.2.1.1.2.2

Rewrite the expression.

$x = 2 \cdot 2$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

Multiply $2$ by $2$ .

$x = 4$