Algebra Examples

$3 \sqrt{27} = 9^{2 x}$

Step 1

Rewrite the equation as $9^{2 x} = 3 \sqrt{27}$ .

$9^{2 x} = 3 \sqrt{27}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{27}$ as $27^{\frac{1}{2}}$ .

$9^{2 x} = 3 \cdot 27^{\frac{1}{2}}$

Step 3

Rewrite $27$ as $3^{3}$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Combine $3$ and $\frac{1}{2}$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Add $2$ and $3$ .

$9^{2 x} = 3^{\frac{5}{2}}$

Step 9

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

$3^{2 (2 x)} = 3^{\frac{5}{2}}$

Step 10

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

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

Step 11

Solve for $x$ .

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Step 11.1

Multiply $2$ by $2$ .

$4 \cdot x = \frac{5}{2}$

Step 11.2

Divide each term in $4 \cdot x = \frac{5}{2}$ by $4$ and simplify.

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Step 11.2.1

Divide each term in $4 \cdot x = \frac{5}{2}$ by $4$ .

$\frac{4 \cdot x}{4} = \frac{\frac{5}{2}}{4}$

Step 11.2.2

Simplify the left side.

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Step 11.2.2.1

Cancel the common factor of $4$ .

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Step 11.2.2.1.1

Cancel the common factor.

$\frac{4 \cdot x}{4} = \frac{\frac{5}{2}}{4}$

Step 11.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5}{2}}{4}$

Step 11.2.3

Simplify the right side.

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Step 11.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5}{2} \cdot \frac{1}{4}$

Step 11.2.3.2

Multiply $\frac{5}{2} \cdot \frac{1}{4}$ .

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Step 11.2.3.2.1

Multiply $\frac{5}{2}$ by $\frac{1}{4}$ .

$x = \frac{5}{2 \cdot 4}$

Step 11.2.3.2.2

Multiply $2$ by $4$ .

$x = \frac{5}{8}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{8}$

Decimal Form:

$x = 0.625$