Pre-Algebra Examples

$\sqrt[3]{27 x \sqrt{x}} = 39$

Step 1

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{27 x \sqrt{x}}}^{3} = 39^{3}$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{27 x \sqrt{x}}$ as ${(27 x \sqrt{x})}^{\frac{1}{3}}$ .

${({(27 x \sqrt{x})}^{\frac{1}{3}})}^{3} = 39^{3}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(27 x \sqrt{x})}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1

Multiply the exponents in ${({(27 x \sqrt{x})}^{\frac{1}{3}})}^{3}$ .

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Step 2.2.1.1.1

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

${(27 x \sqrt{x})}^{\frac{1}{3} \cdot 3} = 39^{3}$

Step 2.2.1.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.1.2.1

Cancel the common factor.

${(27 x \sqrt{x})}^{\frac{1}{3} \cdot 3} = 39^{3}$

Step 2.2.1.1.2.2

Rewrite the expression.

${(27 x \sqrt{x})}^{1} = 39^{3}$

Step 2.2.1.2

Simplify.

$27 x \sqrt{x} = 39^{3}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Raise $39$ to the power of $3$ .

$27 x \sqrt{x} = 59319$

Step 3

To remove the radical on the left side of the equation, square both sides of the equation.

${(27 x \sqrt{x})}^{2} = 59319^{2}$

Step 4

Simplify each side of the equation.

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(27 x \cdot x^{\frac{1}{2}})}^{2} = 59319^{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${(27 x \cdot x^{\frac{1}{2}})}^{2}$ .

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Step 4.2.1.1

Multiply $x$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 4.2.1.1.1

Move $x^{\frac{1}{2}}$ .

${(27 (x^{\frac{1}{2}} x))}^{2} = 59319^{2}$

Step 4.2.1.1.2

Multiply $x^{\frac{1}{2}}$ by $x$ .

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Step 4.2.1.1.2.1

Raise $x$ to the power of $1$ .

${(27 (x^{\frac{1}{2}} x^{1}))}^{2} = 59319^{2}$

Step 4.2.1.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(27 x^{\frac{1}{2} + 1})}^{2} = 59319^{2}$

Step 4.2.1.1.3

Write $1$ as a fraction with a common denominator.

${(27 x^{\frac{1}{2} + \frac{2}{2}})}^{2} = 59319^{2}$

Step 4.2.1.1.4

Combine the numerators over the common denominator.

${(27 x^{\frac{1 + 2}{2}})}^{2} = 59319^{2}$

Step 4.2.1.1.5

Add $1$ and $2$ .

${(27 x^{\frac{3}{2}})}^{2} = 59319^{2}$

Step 4.2.1.2

Apply the product rule to $27 x^{\frac{3}{2}}$ .

$27^{2} {(x^{\frac{3}{2}})}^{2} = 59319^{2}$

Step 4.2.1.3

Raise $27$ to the power of $2$ .

$729 {(x^{\frac{3}{2}})}^{2} = 59319^{2}$

Step 4.2.1.4

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

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Step 4.2.1.4.1

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

$729 x^{\frac{3}{2} \cdot 2} = 59319^{2}$

Step 4.2.1.4.2

Cancel the common factor of $2$ .

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Step 4.2.1.4.2.1

Cancel the common factor.

$729 x^{\frac{3}{2} \cdot 2} = 59319^{2}$

Step 4.2.1.4.2.2

Rewrite the expression.

$729 x^{3} = 59319^{2}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Raise $59319$ to the power of $2$ .

$729 x^{3} = 3518743761$

Step 5

Solve for $x$ .

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Step 5.1

Subtract $3518743761$ from both sides of the equation.

$729 x^{3} - 3518743761 = 0$

Step 5.2

Factor the left side of the equation.

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Step 5.2.1

Factor $729$ out of $729 x^{3} - 3518743761$ .

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Step 5.2.1.1

Factor $729$ out of $729 x^{3}$ .

$729 (x^{3}) - 3518743761 = 0$

Step 5.2.1.2

Factor $729$ out of $- 3518743761$ .

$729 x^{3} + 729 \cdot - 4826809 = 0$

Step 5.2.1.3

Factor $729$ out of $729 x^{3} + 729 \cdot - 4826809$ .

$729 (x^{3} - 4826809) = 0$

Step 5.2.2

Rewrite $4826809$ as $169^{3}$ .

$729 (x^{3} - 169^{3}) = 0$

Step 5.2.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 169$ .

$729 ((x - 169) (x^{2} + x \cdot 169 + 169^{2})) = 0$

Step 5.2.4

Factor.

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Step 5.2.4.1

Simplify.

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Step 5.2.4.1.1

Move $169$ to the left of $x$ .

$729 ((x - 169) (x^{2} + 169 x + 169^{2})) = 0$

Step 5.2.4.1.2

Raise $169$ to the power of $2$ .

$729 ((x - 169) (x^{2} + 169 x + 28561)) = 0$

Step 5.2.4.2

Remove unnecessary parentheses.

$729 (x - 169) (x^{2} + 169 x + 28561) = 0$

Step 5.3

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

$x - 169 = 0$

$x^{2} + 169 x + 28561 = 0$

Step 5.4

Set $x - 169$ equal to $0$ and solve for $x$ .

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Step 5.4.1

Set $x - 169$ equal to $0$ .

$x - 169 = 0$

Step 5.4.2

Add $169$ to both sides of the equation.

$x = 169$

Step 5.5

Set $x^{2} + 169 x + 28561$ equal to $0$ and solve for $x$ .

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Step 5.5.1

Set $x^{2} + 169 x + 28561$ equal to $0$ .

$x^{2} + 169 x + 28561 = 0$

Step 5.5.2

Solve $x^{2} + 169 x + 28561 = 0$ for $x$ .

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Step 5.5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5.5.2.2

Substitute the values $a = 1$ , $b = 169$ , and $c = 28561$ into the quadratic formula and solve for $x$ .

$\frac{- 169 \pm \sqrt{169^{2} - 4 \cdot (1 \cdot 28561)}}{2 \cdot 1}$

Step 5.5.2.3

Simplify.

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Step 5.5.2.3.1

Simplify the numerator.

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Step 5.5.2.3.1.1

Raise $169$ to the power of $2$ .

$x = \frac{- 169 \pm \sqrt{28561 - 4 \cdot 1 \cdot 28561}}{2 \cdot 1}$

Step 5.5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 28561$ .

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Step 5.5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 169 \pm \sqrt{28561 - 4 \cdot 28561}}{2 \cdot 1}$

Step 5.5.2.3.1.2.2

Multiply $- 4$ by $28561$ .

$x = \frac{- 169 \pm \sqrt{28561 - 114244}}{2 \cdot 1}$

Step 5.5.2.3.1.3

Subtract $114244$ from $28561$ .

$x = \frac{- 169 \pm \sqrt{- 85683}}{2 \cdot 1}$

Step 5.5.2.3.1.4

Rewrite $- 85683$ as $- 1 (85683)$ .

$x = \frac{- 169 \pm \sqrt{- 1 \cdot 85683}}{2 \cdot 1}$

Step 5.5.2.3.1.5

Rewrite $\sqrt{- 1 (85683)}$ as $\sqrt{- 1} \cdot \sqrt{85683}$ .

$x = \frac{- 169 \pm \sqrt{- 1} \cdot \sqrt{85683}}{2 \cdot 1}$

Step 5.5.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 169 \pm i \cdot \sqrt{85683}}{2 \cdot 1}$

Step 5.5.2.3.1.7

Rewrite $85683$ as $169^{2} \cdot 3$ .

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Step 5.5.2.3.1.7.1

Factor $28561$ out of $85683$ .

$x = \frac{- 169 \pm i \cdot \sqrt{28561 (3)}}{2 \cdot 1}$

Step 5.5.2.3.1.7.2

Rewrite $28561$ as $169^{2}$ .

$x = \frac{- 169 \pm i \cdot \sqrt{169^{2} \cdot 3}}{2 \cdot 1}$

Step 5.5.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 169 \pm i \cdot (169 \sqrt{3})}{2 \cdot 1}$

Step 5.5.2.3.1.9

Move $169$ to the left of $i$ .

$x = \frac{- 169 \pm 169 i \sqrt{3}}{2 \cdot 1}$

Step 5.5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 169 \pm 169 i \sqrt{3}}{2}$

Step 5.5.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 5.5.2.4.1

Simplify the numerator.

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Step 5.5.2.4.1.1

Raise $169$ to the power of $2$ .

$x = \frac{- 169 \pm \sqrt{28561 - 4 \cdot 1 \cdot 28561}}{2 \cdot 1}$

Step 5.5.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 28561$ .

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Step 5.5.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 169 \pm \sqrt{28561 - 4 \cdot 28561}}{2 \cdot 1}$

Step 5.5.2.4.1.2.2

Multiply $- 4$ by $28561$ .

$x = \frac{- 169 \pm \sqrt{28561 - 114244}}{2 \cdot 1}$

Step 5.5.2.4.1.3

Subtract $114244$ from $28561$ .

$x = \frac{- 169 \pm \sqrt{- 85683}}{2 \cdot 1}$

Step 5.5.2.4.1.4

Rewrite $- 85683$ as $- 1 (85683)$ .

$x = \frac{- 169 \pm \sqrt{- 1 \cdot 85683}}{2 \cdot 1}$

Step 5.5.2.4.1.5

Rewrite $\sqrt{- 1 (85683)}$ as $\sqrt{- 1} \cdot \sqrt{85683}$ .

$x = \frac{- 169 \pm \sqrt{- 1} \cdot \sqrt{85683}}{2 \cdot 1}$

Step 5.5.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 169 \pm i \cdot \sqrt{85683}}{2 \cdot 1}$

Step 5.5.2.4.1.7

Rewrite $85683$ as $169^{2} \cdot 3$ .

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Step 5.5.2.4.1.7.1

Factor $28561$ out of $85683$ .

$x = \frac{- 169 \pm i \cdot \sqrt{28561 (3)}}{2 \cdot 1}$

Step 5.5.2.4.1.7.2

Rewrite $28561$ as $169^{2}$ .

$x = \frac{- 169 \pm i \cdot \sqrt{169^{2} \cdot 3}}{2 \cdot 1}$

Step 5.5.2.4.1.8

Pull terms out from under the radical.

$x = \frac{- 169 \pm i \cdot (169 \sqrt{3})}{2 \cdot 1}$

Step 5.5.2.4.1.9

Move $169$ to the left of $i$ .

$x = \frac{- 169 \pm 169 i \sqrt{3}}{2 \cdot 1}$

Step 5.5.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 169 \pm 169 i \sqrt{3}}{2}$

Step 5.5.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 169 + 169 i \sqrt{3}}{2}$

Step 5.5.2.4.4

Rewrite $- 169$ as $- 1 (169)$ .

$x = \frac{- 1 \cdot 169 + 169 i \sqrt{3}}{2}$

Step 5.5.2.4.5

Factor $- 1$ out of $169 i \sqrt{3}$ .

$x = \frac{- 1 \cdot 169 - (- 169 i \sqrt{3})}{2}$

Step 5.5.2.4.6

Factor $- 1$ out of $- 1 (169) - (- 169 i \sqrt{3})$ .

$x = \frac{- 1 (169 - 169 i \sqrt{3})}{2}$

Step 5.5.2.4.7

Move the negative in front of the fraction.

$x = - \frac{169 - 169 i \sqrt{3}}{2}$

Step 5.5.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 5.5.2.5.1

Simplify the numerator.

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Step 5.5.2.5.1.1

Raise $169$ to the power of $2$ .

$x = \frac{- 169 \pm \sqrt{28561 - 4 \cdot 1 \cdot 28561}}{2 \cdot 1}$

Step 5.5.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 28561$ .

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Step 5.5.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 169 \pm \sqrt{28561 - 4 \cdot 28561}}{2 \cdot 1}$

Step 5.5.2.5.1.2.2

Multiply $- 4$ by $28561$ .

$x = \frac{- 169 \pm \sqrt{28561 - 114244}}{2 \cdot 1}$

Step 5.5.2.5.1.3

Subtract $114244$ from $28561$ .

$x = \frac{- 169 \pm \sqrt{- 85683}}{2 \cdot 1}$

Step 5.5.2.5.1.4

Rewrite $- 85683$ as $- 1 (85683)$ .

$x = \frac{- 169 \pm \sqrt{- 1 \cdot 85683}}{2 \cdot 1}$

Step 5.5.2.5.1.5

Rewrite $\sqrt{- 1 (85683)}$ as $\sqrt{- 1} \cdot \sqrt{85683}$ .

$x = \frac{- 169 \pm \sqrt{- 1} \cdot \sqrt{85683}}{2 \cdot 1}$

Step 5.5.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 169 \pm i \cdot \sqrt{85683}}{2 \cdot 1}$

Step 5.5.2.5.1.7

Rewrite $85683$ as $169^{2} \cdot 3$ .

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Step 5.5.2.5.1.7.1

Factor $28561$ out of $85683$ .

$x = \frac{- 169 \pm i \cdot \sqrt{28561 (3)}}{2 \cdot 1}$

Step 5.5.2.5.1.7.2

Rewrite $28561$ as $169^{2}$ .

$x = \frac{- 169 \pm i \cdot \sqrt{169^{2} \cdot 3}}{2 \cdot 1}$

Step 5.5.2.5.1.8

Pull terms out from under the radical.

$x = \frac{- 169 \pm i \cdot (169 \sqrt{3})}{2 \cdot 1}$

Step 5.5.2.5.1.9

Move $169$ to the left of $i$ .

$x = \frac{- 169 \pm 169 i \sqrt{3}}{2 \cdot 1}$

Step 5.5.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 169 \pm 169 i \sqrt{3}}{2}$

Step 5.5.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 169 - 169 i \sqrt{3}}{2}$

Step 5.5.2.5.4

Rewrite $- 169$ as $- 1 (169)$ .

$x = \frac{- 1 \cdot 169 - 169 i \sqrt{3}}{2}$

Step 5.5.2.5.5

Factor $- 1$ out of $- 169 i \sqrt{3}$ .

$x = \frac{- 1 \cdot 169 - (169 i \sqrt{3})}{2}$

Step 5.5.2.5.6

Factor $- 1$ out of $- 1 (169) - (169 i \sqrt{3})$ .

$x = \frac{- 1 (169 + 169 i \sqrt{3})}{2}$

Step 5.5.2.5.7

Move the negative in front of the fraction.

$x = - \frac{169 + 169 i \sqrt{3}}{2}$

Step 5.5.2.6

The final answer is the combination of both solutions.

$x = - \frac{169 - 169 i \sqrt{3}}{2}, - \frac{169 + 169 i \sqrt{3}}{2}$

Step 5.6

The final solution is all the values that make $729 (x - 169) (x^{2} + 169 x + 28561) = 0$ true.

$x = 169, - \frac{169 - 169 i \sqrt{3}}{2}, - \frac{169 + 169 i \sqrt{3}}{2}$