Trigonometry Examples

${(2 x + 3)}^{\frac{3}{2}} = 64$

Step 1

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

${({(2 x + 3)}^{\frac{3}{2}})}^{\frac{2}{3}} = 64^{\frac{2}{3}}$

Step 2

Simplify the exponent.

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Step 2.1

Simplify the left side.

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Step 2.1.1

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

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Step 2.1.1.1

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

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Step 2.1.1.1.1

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

${(2 x + 3)}^{\frac{3}{2} \cdot \frac{2}{3}} = 64^{\frac{2}{3}}$

Step 2.1.1.1.2

Cancel the common factor of $3$ .

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Step 2.1.1.1.2.1

Cancel the common factor.

${(2 x + 3)}^{\frac{3}{2} \cdot \frac{2}{3}} = 64^{\frac{2}{3}}$

Step 2.1.1.1.2.2

Rewrite the expression.

${(2 x + 3)}^{\frac{1}{2} \cdot 2} = 64^{\frac{2}{3}}$

Step 2.1.1.1.3

Cancel the common factor of $2$ .

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Step 2.1.1.1.3.1

Cancel the common factor.

${(2 x + 3)}^{\frac{1}{2} \cdot 2} = 64^{\frac{2}{3}}$

Step 2.1.1.1.3.2

Rewrite the expression.

${(2 x + 3)}^{1} = 64^{\frac{2}{3}}$

Step 2.1.1.2

Simplify.

$2 x + 3 = 64^{\frac{2}{3}}$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $64^{\frac{2}{3}}$ .

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Step 2.2.1.1

Simplify the expression.

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Step 2.2.1.1.1

Rewrite $64$ as $4^{3}$ .

$2 x + 3 = {(4^{3})}^{\frac{2}{3}}$

Step 2.2.1.1.2

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

$2 x + 3 = 4^{3 (\frac{2}{3})}$

Step 2.2.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.2.1

Cancel the common factor.

$2 x + 3 = 4^{3 (\frac{2}{3})}$

Step 2.2.1.2.2

Rewrite the expression.

$2 x + 3 = 4^{2}$

Step 2.2.1.3

Raise $4$ to the power of $2$ .

$2 x + 3 = 16$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1.1

Subtract $3$ from both sides of the equation.

$2 x = 16 - 3$

Step 3.1.2

Subtract $3$ from $16$ .

$2 x = 13$

Step 3.2

Divide each term in $2 x = 13$ by $2$ and simplify.

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Step 3.2.1

Divide each term in $2 x = 13$ by $2$ .

$\frac{2 x}{2} = \frac{13}{2}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{13}{2}$

Step 3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{13}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{13}{2}$

Decimal Form:

$x = 6.5$

Mixed Number Form:

$x = 6 \frac{1}{2}$