Pre-Algebra Examples

$6 x^{\frac{2}{3}} - 41 x^{\frac{1}{3}} - 7 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

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

$6 {(x^{\frac{1}{3}})}^{2} - 41 x^{\frac{1}{3}} - 7 = 0$

Step 1.2

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

$6 u^{2} - 41 u - 7 = 0$

Step 1.3

Factor by grouping.

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Step 1.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 6 \cdot - 7 = - 42$ and whose sum is $b = - 41$ .

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Step 1.3.1.1

Factor $- 41$ out of $- 41 u$ .

$6 u^{2} - 41 u - 7 = 0$

Step 1.3.1.2

Rewrite $- 41$ as $1$ plus $- 42$

$6 u^{2} + (1 - 42) u - 7 = 0$

Step 1.3.1.3

Apply the distributive property.

$6 u^{2} + 1 u - 42 u - 7 = 0$

Step 1.3.1.4

Multiply $u$ by $1$ .

$6 u^{2} + u - 42 u - 7 = 0$

Step 1.3.2

Factor out the greatest common factor from each group.

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Step 1.3.2.1

Group the first two terms and the last two terms.

$(6 u^{2} + u) - 42 u - 7 = 0$

Step 1.3.2.2

Factor out the greatest common factor (GCF) from each group.

$u (6 u + 1) - 7 (6 u + 1) = 0$

Step 1.3.3

Factor the polynomial by factoring out the greatest common factor, $6 u + 1$ .

$(6 u + 1) (u - 7) = 0$

Step 1.4

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

$(6 x^{\frac{1}{3}} + 1) (x^{\frac{1}{3}} - 7) = 0$

Step 2

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

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

$x^{\frac{1}{3}} - 7 = 0$

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Subtract $1$ from both sides of the equation.

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

Step 3.2.2

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

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

Step 3.2.3

Simplify the exponent.

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Step 3.2.3.1

Simplify the left side.

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Step 3.2.3.1.1

Simplify ${(6 x^{\frac{1}{3}})}^{3}$ .

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Step 3.2.3.1.1.1

Apply the product rule to $6 x^{\frac{1}{3}}$ .

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

Step 3.2.3.1.1.2

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

$216 {(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 3.2.3.1.1.3

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

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Step 3.2.3.1.1.3.1

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

$216 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 3.2.3.1.1.3.2

Cancel the common factor of $3$ .

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Step 3.2.3.1.1.3.2.1

Cancel the common factor.

$216 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 3.2.3.1.1.3.2.2

Rewrite the expression.

$216 x^{1} = {(- 1)}^{3}$

Step 3.2.3.1.1.4

Simplify.

$216 x = {(- 1)}^{3}$

Step 3.2.3.2

Simplify the right side.

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Step 3.2.3.2.1

Raise $- 1$ to the power of $3$ .

$216 x = - 1$

Step 3.2.4

Divide each term in $216 x = - 1$ by $216$ and simplify.

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Step 3.2.4.1

Divide each term in $216 x = - 1$ by $216$ .

$\frac{216 x}{216} = \frac{- 1}{216}$

Step 3.2.4.2

Simplify the left side.

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Step 3.2.4.2.1

Cancel the common factor of $216$ .

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Step 3.2.4.2.1.1

Cancel the common factor.

$\frac{216 x}{216} = \frac{- 1}{216}$

Step 3.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{216}$

Step 3.2.4.3

Simplify the right side.

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Step 3.2.4.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{216}$

Step 4

Set $x^{\frac{1}{3}} - 7$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{\frac{1}{3}} - 7$ equal to $0$ .

$x^{\frac{1}{3}} - 7 = 0$

Step 4.2

Solve $x^{\frac{1}{3}} - 7 = 0$ for $x$ .

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Step 4.2.1

Add $7$ to both sides of the equation.

$x^{\frac{1}{3}} = 7$

Step 4.2.2

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

${(x^{\frac{1}{3}})}^{3} = 7^{3}$

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

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Step 4.2.3.1.1.1

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

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Step 4.2.3.1.1.1.1

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

$x^{\frac{1}{3} \cdot 3} = 7^{3}$

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.

$x^{\frac{1}{3} \cdot 3} = 7^{3}$

Step 4.2.3.1.1.1.2.2

Rewrite the expression.

$x^{1} = 7^{3}$

Step 4.2.3.1.1.2

Simplify.

$x = 7^{3}$

Step 4.2.3.2

Simplify the right side.

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Step 4.2.3.2.1

Raise $7$ to the power of $3$ .

$x = 343$

Step 5

The final solution is all the values that make $(6 x^{\frac{1}{3}} + 1) (x^{\frac{1}{3}} - 7) = 0$ true.

$x = - \frac{1}{216}, 343$