Algebra Examples

$x^{\frac{2}{3}} - 2 x^{\frac{1}{3}} - 15 = 0$

Step 1

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

${(x^{\frac{1}{3}})}^{2} - 2 x^{\frac{1}{3}} - 15 = 0$

Step 2

Let $u = x^{\frac{1}{3}}$ . Substitute $u$ for all occurrences of $x^{\frac{1}{3}}$ .

$u^{2} - 2 u - 15 = 0$

Step 3

Factor $u^{2} - 2 u - 15$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Step 3.2

Write the factored form using these integers.

$(u - 5) (u + 3) = 0$

Step 4

Replace all occurrences of $u$ with $x^{\frac{1}{3}}$ .

$(x^{\frac{1}{3}} - 5) (x^{\frac{1}{3}} + 3) = 0$

Step 5

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

$x^{\frac{1}{3}} - 5 = 0$

$x^{\frac{1}{3}} + 3 = 0$

Step 6

Set $x^{\frac{1}{3}} - 5$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x^{\frac{1}{3}} - 5$ equal to $0$ .

$x^{\frac{1}{3}} - 5 = 0$

Step 6.2

Solve $x^{\frac{1}{3}} - 5 = 0$ for $x$ .

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Step 6.2.1

Add $5$ to both sides of the equation.

$x^{\frac{1}{3}} = 5$

Step 6.2.2

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

${(x^{\frac{1}{3}})}^{3} = 5^{3}$

Step 6.2.3

Simplify the exponent.

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Step 6.2.3.1

Simplify the left side.

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Step 6.2.3.1.1

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

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Step 6.2.3.1.1.1

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

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Step 6.2.3.1.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = 5^{3}$

Step 6.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 6.2.3.1.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 5^{3}$

Step 6.2.3.1.1.1.2.2

Rewrite the expression.

$x^{1} = 5^{3}$

Step 6.2.3.1.1.2

Simplify.

$x = 5^{3}$

Step 6.2.3.2

Simplify the right side.

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Step 6.2.3.2.1

Raise $5$ to the power of $3$ .

$x = 125$

Step 7

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

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Step 7.1

Set $x^{\frac{1}{3}} + 3$ equal to $0$ .

$x^{\frac{1}{3}} + 3 = 0$

Step 7.2

Solve $x^{\frac{1}{3}} + 3 = 0$ for $x$ .

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Step 7.2.1

Subtract $3$ from both sides of the equation.

$x^{\frac{1}{3}} = - 3$

Step 7.2.2

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

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

Step 7.2.3

Simplify the exponent.

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Step 7.2.3.1

Simplify the left side.

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Step 7.2.3.1.1

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

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Step 7.2.3.1.1.1

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

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Step 7.2.3.1.1.1.1

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

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

Step 7.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 7.2.3.1.1.1.2.1

Cancel the common factor.

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

Step 7.2.3.1.1.1.2.2

Rewrite the expression.

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

Step 7.2.3.1.1.2

Simplify.

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

Step 7.2.3.2

Simplify the right side.

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Step 7.2.3.2.1

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

$x = - 27$

Step 8

The final solution is all the values that make $(x^{\frac{1}{3}} - 5) (x^{\frac{1}{3}} + 3) = 0$ true.

$x = 125, - 27$