Algebra Examples

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

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.

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Step 2.1

Simplify the left side.

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Step 2.1.1

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

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Step 2.1.1.1

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

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Step 2.1.1.1.1

Cancel the common factor.

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

Step 2.1.1.1.2

Rewrite the expression.

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

Step 2.1.1.2

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

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Step 2.1.1.2.1

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

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

Step 2.1.1.2.2

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

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

Step 2.2

Simplify the right side.

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Step 2.2.1

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

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Step 2.2.1.1

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

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Step 2.2.1.1.1

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

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

Step 2.2.1.1.2

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

Combine the numerators over the common denominator.

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

Step 2.2.1.4

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.

$x = 3$