Trigonometry Examples

${(x + 2)}^{\frac{2}{3}} = 27$

Step 1

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

${({(x + 2)}^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 27^{\frac{3}{2}}$

Step 2

Simplify the left side.

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Step 2.1

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

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Step 2.1.1

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

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Step 2.1.1.1

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

${(x + 2)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 27^{\frac{3}{2}}$

Step 2.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.1.2.1

Cancel the common factor.

${(x + 2)}^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 27^{\frac{3}{2}}$

Step 2.1.1.2.2

Rewrite the expression.

${(x + 2)}^{\frac{1}{3} \cdot 3} = \pm 27^{\frac{3}{2}}$

Step 2.1.1.3

Cancel the common factor of $3$ .

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Step 2.1.1.3.1

Cancel the common factor.

${(x + 2)}^{\frac{1}{3} \cdot 3} = \pm 27^{\frac{3}{2}}$

Step 2.1.1.3.2

Rewrite the expression.

${(x + 2)}^{1} = \pm 27^{\frac{3}{2}}$

Step 2.1.2

Simplify.

$x + 2 = \pm 27^{\frac{3}{2}}$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.1

First, use the positive value of the $\pm$ to find the first solution.

$x + 2 = 27^{\frac{3}{2}}$

Step 3.2

Subtract $2$ from both sides of the equation.

$x = 27^{\frac{3}{2}} - 2$

Step 3.3

Next, use the negative value of the $\pm$ to find the second solution.

$x + 2 = - 27^{\frac{3}{2}}$

Step 3.4

Subtract $2$ from both sides of the equation.

$x = - 27^{\frac{3}{2}} - 2$

Step 3.5

The complete solution is the result of both the positive and negative portions of the solution.

$x = 27^{\frac{3}{2}} - 2, - 27^{\frac{3}{2}} - 2$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = 27^{\frac{3}{2}} - 2, - 27^{\frac{3}{2}} - 2$

Decimal Form:

$x = 138.29611541 \dots, - 142.29611541 \dots$