Precalculus Examples

${(x - 3)}^{\frac{2}{3}} = 36$

Step 1

Subtract $36$ from both sides of the equation.

${(x - 3)}^{\frac{2}{3}} - 36 = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Rewrite ${(x - 3)}^{\frac{2}{3}}$ as ${({(x - 3)}^{\frac{1}{3}})}^{2}$ .

${({(x - 3)}^{\frac{1}{3}})}^{2} - 36 = 0$

Step 2.2

Rewrite $36$ as $6^{2}$ .

${({(x - 3)}^{\frac{1}{3}})}^{2} - 6^{2} = 0$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x - 3)}^{\frac{1}{3}}$ and $b = 6$ .

$({(x - 3)}^{\frac{1}{3}} + 6) ({(x - 3)}^{\frac{1}{3}} - 6) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x - 3)}^{\frac{1}{3}} + 6 = 0$

${(x - 3)}^{\frac{1}{3}} - 6 = 0$

Step 4

Set ${(x - 3)}^{\frac{1}{3}} + 6$ equal to $0$ and solve for $x$ .

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Step 4.1

Set ${(x - 3)}^{\frac{1}{3}} + 6$ equal to $0$ .

${(x - 3)}^{\frac{1}{3}} + 6 = 0$

Step 4.2

Solve ${(x - 3)}^{\frac{1}{3}} + 6 = 0$ for $x$ .

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Step 4.2.1

Subtract $6$ from both sides of the equation.

${(x - 3)}^{\frac{1}{3}} = - 6$

Step 4.2.2

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${({(x - 3)}^{\frac{1}{3}})}^{3} = {(- 6)}^{3}$

Step 4.2.3

Simplify the exponent.

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Step 4.2.3.1

Simplify the left side.

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Step 4.2.3.1.1

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

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Step 4.2.3.1.1.1

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

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Step 4.2.3.1.1.1.1

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

${(x - 3)}^{\frac{1}{3} \cdot 3} = {(- 6)}^{3}$

Step 4.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 4.2.3.1.1.1.2.1

Cancel the common factor.

${(x - 3)}^{\frac{1}{3} \cdot 3} = {(- 6)}^{3}$

Step 4.2.3.1.1.1.2.2

Rewrite the expression.

${(x - 3)}^{1} = {(- 6)}^{3}$

Step 4.2.3.1.1.2

Simplify.

$x - 3 = {(- 6)}^{3}$

Step 4.2.3.2

Simplify the right side.

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Step 4.2.3.2.1

Raise $- 6$ to the power of $3$ .

$x - 3 = - 216$

Step 4.2.4

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

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Step 4.2.4.1

Add $3$ to both sides of the equation.

$x = - 216 + 3$

Step 4.2.4.2

Add $- 216$ and $3$ .

$x = - 213$

Step 5

Set ${(x - 3)}^{\frac{1}{3}} - 6$ equal to $0$ and solve for $x$ .

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Step 5.1

Set ${(x - 3)}^{\frac{1}{3}} - 6$ equal to $0$ .

${(x - 3)}^{\frac{1}{3}} - 6 = 0$

Step 5.2

Solve ${(x - 3)}^{\frac{1}{3}} - 6 = 0$ for $x$ .

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Step 5.2.1

Add $6$ to both sides of the equation.

${(x - 3)}^{\frac{1}{3}} = 6$

Step 5.2.2

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

${({(x - 3)}^{\frac{1}{3}})}^{3} = 6^{3}$

Step 5.2.3

Simplify the exponent.

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Step 5.2.3.1

Simplify the left side.

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Step 5.2.3.1.1

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

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Step 5.2.3.1.1.1

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

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Step 5.2.3.1.1.1.1

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

${(x - 3)}^{\frac{1}{3} \cdot 3} = 6^{3}$

Step 5.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 5.2.3.1.1.1.2.1

Cancel the common factor.

${(x - 3)}^{\frac{1}{3} \cdot 3} = 6^{3}$

Step 5.2.3.1.1.1.2.2

Rewrite the expression.

${(x - 3)}^{1} = 6^{3}$

Step 5.2.3.1.1.2

Simplify.

$x - 3 = 6^{3}$

Step 5.2.3.2

Simplify the right side.

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Step 5.2.3.2.1

Raise $6$ to the power of $3$ .

$x - 3 = 216$

Step 5.2.4

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

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Step 5.2.4.1

Add $3$ to both sides of the equation.

$x = 216 + 3$

Step 5.2.4.2

Add $216$ and $3$ .

$x = 219$

Step 6

The final solution is all the values that make $({(x - 3)}^{\frac{1}{3}} + 6) ({(x - 3)}^{\frac{1}{3}} - 6) = 0$ true.

$x = - 213, 219$