Algebra Examples

$\sqrt{5^{2} + 12^{2}} = \sqrt[3]{64 a^{3}}$

Step 1

Rewrite the equation as $\sqrt[3]{64 a^{3}} = \sqrt{5^{2} + 12^{2}}$ .

$\sqrt[3]{64 a^{3}} = \sqrt{5^{2} + 12^{2}}$

Step 2

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

${\sqrt[3]{64 a^{3}}}^{3} = {\sqrt{5^{2} + 12^{2}}}^{3}$

Step 3

Simplify each side of the equation.

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Step 3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{64 a^{3}}$ as ${(64 a^{3})}^{\frac{1}{3}}$ .

${({(64 a^{3})}^{\frac{1}{3}})}^{3} = {\sqrt{5^{2} + 12^{2}}}^{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify ${({(64 a^{3})}^{\frac{1}{3}})}^{3}$ .

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Step 3.2.1.1

Multiply the exponents in ${({(64 a^{3})}^{\frac{1}{3}})}^{3}$ .

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Step 3.2.1.1.1

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

${(64 a^{3})}^{\frac{1}{3} \cdot 3} = {\sqrt{5^{2} + 12^{2}}}^{3}$

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

${(64 a^{3})}^{\frac{1}{3} \cdot 3} = {\sqrt{5^{2} + 12^{2}}}^{3}$

Step 3.2.1.1.2.2

Rewrite the expression.

${(64 a^{3})}^{1} = {\sqrt{5^{2} + 12^{2}}}^{3}$

Step 3.2.1.2

Simplify.

$64 a^{3} = {\sqrt{5^{2} + 12^{2}}}^{3}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify ${\sqrt{5^{2} + 12^{2}}}^{3}$ .

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Step 3.3.1.1

Raise $5$ to the power of $2$ .

$64 a^{3} = {\sqrt{25 + 12^{2}}}^{3}$

Step 3.3.1.2

Raise $12$ to the power of $2$ .

$64 a^{3} = {\sqrt{25 + 144}}^{3}$

Step 3.3.1.3

Add $25$ and $144$ .

$64 a^{3} = {\sqrt{169}}^{3}$

Step 3.3.1.4

Rewrite $169$ as $13^{2}$ .

$64 a^{3} = {\sqrt{13^{2}}}^{3}$

Step 3.3.1.5

Pull terms out from under the radical, assuming positive real numbers.

$64 a^{3} = 13^{3}$

Step 3.3.1.6

Raise $13$ to the power of $3$ .

$64 a^{3} = 2197$

Step 4

Solve for $a$ .

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Step 4.1

Subtract $2197$ from both sides of the equation.

$64 a^{3} - 2197 = 0$

Step 4.2

Factor the left side of the equation.

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Step 4.2.1

Rewrite $64 a^{3}$ as ${(4 a)}^{3}$ .

${(4 a)}^{3} - 2197 = 0$

Step 4.2.2

Rewrite $2197$ as $13^{3}$ .

${(4 a)}^{3} - 13^{3} = 0$

Step 4.2.3

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 $a = 4 a$ and $b = 13$ .

$(4 a - 13) ({(4 a)}^{2} + 4 a \cdot 13 + 13^{2}) = 0$

Step 4.2.4

Simplify.

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Step 4.2.4.1

Apply the product rule to $4 a$ .

$(4 a - 13) (4^{2} a^{2} + 4 a \cdot 13 + 13^{2}) = 0$

Step 4.2.4.2

Raise $4$ to the power of $2$ .

$(4 a - 13) (16 a^{2} + 4 a \cdot 13 + 13^{2}) = 0$

Step 4.2.4.3

Multiply $13$ by $4$ .

$(4 a - 13) (16 a^{2} + 52 a + 13^{2}) = 0$

Step 4.2.4.4

Raise $13$ to the power of $2$ .

$(4 a - 13) (16 a^{2} + 52 a + 169) = 0$

Step 4.3

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

$4 a - 13 = 0$

$16 a^{2} + 52 a + 169 = 0$

Step 4.4

Set $4 a - 13$ equal to $0$ and solve for $a$ .

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Step 4.4.1

Set $4 a - 13$ equal to $0$ .

$4 a - 13 = 0$

Step 4.4.2

Solve $4 a - 13 = 0$ for $a$ .

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Step 4.4.2.1

Add $13$ to both sides of the equation.

$4 a = 13$

Step 4.4.2.2

Divide each term in $4 a = 13$ by $4$ and simplify.

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Step 4.4.2.2.1

Divide each term in $4 a = 13$ by $4$ .

$\frac{4 a}{4} = \frac{13}{4}$

Step 4.4.2.2.2

Simplify the left side.

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Step 4.4.2.2.2.1

Cancel the common factor of $4$ .

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Step 4.4.2.2.2.1.1

Cancel the common factor.

$\frac{4 a}{4} = \frac{13}{4}$

Step 4.4.2.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{13}{4}$

Step 4.5

Set $16 a^{2} + 52 a + 169$ equal to $0$ and solve for $a$ .

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Step 4.5.1

Set $16 a^{2} + 52 a + 169$ equal to $0$ .

$16 a^{2} + 52 a + 169 = 0$

Step 4.5.2

Solve $16 a^{2} + 52 a + 169 = 0$ for $a$ .

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Step 4.5.2.1

Use the quadratic formula to find the solutions.

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

Step 4.5.2.2

Substitute the values $a = 16$ , $b = 52$ , and $c = 169$ into the quadratic formula and solve for $a$ .

$\frac{- 52 \pm \sqrt{52^{2} - 4 \cdot (16 \cdot 169)}}{2 \cdot 16}$

Step 4.5.2.3

Simplify.

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Step 4.5.2.3.1

Simplify the numerator.

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Step 4.5.2.3.1.1

Raise $52$ to the power of $2$ .

$a = \frac{- 52 \pm \sqrt{2704 - 4 \cdot 16 \cdot 169}}{2 \cdot 16}$

Step 4.5.2.3.1.2

Multiply $- 4 \cdot 16 \cdot 169$ .

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Step 4.5.2.3.1.2.1

Multiply $- 4$ by $16$ .

$a = \frac{- 52 \pm \sqrt{2704 - 64 \cdot 169}}{2 \cdot 16}$

Step 4.5.2.3.1.2.2

Multiply $- 64$ by $169$ .

$a = \frac{- 52 \pm \sqrt{2704 - 10816}}{2 \cdot 16}$

Step 4.5.2.3.1.3

Subtract $10816$ from $2704$ .

$a = \frac{- 52 \pm \sqrt{- 8112}}{2 \cdot 16}$

Step 4.5.2.3.1.4

Rewrite $- 8112$ as $- 1 (8112)$ .

$a = \frac{- 52 \pm \sqrt{- 1 \cdot 8112}}{2 \cdot 16}$

Step 4.5.2.3.1.5

Rewrite $\sqrt{- 1 (8112)}$ as $\sqrt{- 1} \cdot \sqrt{8112}$ .

$a = \frac{- 52 \pm \sqrt{- 1} \cdot \sqrt{8112}}{2 \cdot 16}$

Step 4.5.2.3.1.6

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

$a = \frac{- 52 \pm i \cdot \sqrt{8112}}{2 \cdot 16}$

Step 4.5.2.3.1.7

Rewrite $8112$ as $52^{2} \cdot 3$ .

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Step 4.5.2.3.1.7.1

Factor $2704$ out of $8112$ .

$a = \frac{- 52 \pm i \cdot \sqrt{2704 (3)}}{2 \cdot 16}$

Step 4.5.2.3.1.7.2

Rewrite $2704$ as $52^{2}$ .

$a = \frac{- 52 \pm i \cdot \sqrt{52^{2} \cdot 3}}{2 \cdot 16}$

Step 4.5.2.3.1.8

Pull terms out from under the radical.

$a = \frac{- 52 \pm i \cdot (52 \sqrt{3})}{2 \cdot 16}$

Step 4.5.2.3.1.9

Move $52$ to the left of $i$ .

$a = \frac{- 52 \pm 52 i \sqrt{3}}{2 \cdot 16}$

Step 4.5.2.3.2

Multiply $2$ by $16$ .

$a = \frac{- 52 \pm 52 i \sqrt{3}}{32}$

Step 4.5.2.3.3

Simplify $\frac{- 52 \pm 52 i \sqrt{3}}{32}$ .

$a = \frac{- 13 \pm 13 i \sqrt{3}}{8}$

Step 4.5.2.4

The final answer is the combination of both solutions.

$a = - \frac{13 - 13 i \sqrt{3}}{8}, - \frac{13 + 13 i \sqrt{3}}{8}$

Step 4.6

The final solution is all the values that make $(4 a - 13) (16 a^{2} + 52 a + 169) = 0$ true.

$a = \frac{13}{4}, - \frac{13 - 13 i \sqrt{3}}{8}, - \frac{13 + 13 i \sqrt{3}}{8}$