Algebra Examples

$r^{\frac{6}{5}} = 64$

Step 1

Subtract $64$ from both sides of the equation.

$r^{\frac{6}{5}} - 64 = 0$

Step 2

Rewrite $r^{\frac{6}{5}}$ as ${(r^{\frac{2}{5}})}^{3}$ .

${(r^{\frac{2}{5}})}^{3} - 64 = 0$

Step 3

Rewrite $64$ as $4^{3}$ .

${(r^{\frac{2}{5}})}^{3} - 4^{3} = 0$

Step 4

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 $a = r^{\frac{2}{5}}$ and $b = 4$ .

$(r^{\frac{2}{5}} - 4) ({(r^{\frac{2}{5}})}^{2} + r^{\frac{2}{5}} \cdot 4 + 4^{2}) = 0$

Step 5

Simplify.

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Step 5.1

Rewrite $r^{\frac{2}{5}}$ as ${(r^{\frac{1}{5}})}^{2}$ .

$({(r^{\frac{1}{5}})}^{2} - 4) ({(r^{\frac{2}{5}})}^{2} + r^{\frac{2}{5}} \cdot 4 + 4^{2}) = 0$

Step 5.2

Rewrite $4$ as $2^{2}$ .

$({(r^{\frac{1}{5}})}^{2} - 2^{2}) ({(r^{\frac{2}{5}})}^{2} + r^{\frac{2}{5}} \cdot 4 + 4^{2}) = 0$

Step 5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = r^{\frac{1}{5}}$ and $b = 2$ .

$(r^{\frac{1}{5}} + 2) (r^{\frac{1}{5}} - 2) ({(r^{\frac{2}{5}})}^{2} + r^{\frac{2}{5}} \cdot 4 + 4^{2}) = 0$

Step 5.4

Multiply the exponents in ${(r^{\frac{2}{5}})}^{2}$ .

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Step 5.4.1

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

$(r^{\frac{1}{5}} + 2) (r^{\frac{1}{5}} - 2) (r^{\frac{2}{5} \cdot 2} + r^{\frac{2}{5}} \cdot 4 + 4^{2}) = 0$

Step 5.4.2

Multiply $\frac{2}{5} \cdot 2$ .

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Step 5.4.2.1

Combine $\frac{2}{5}$ and $2$ .

$(r^{\frac{1}{5}} + 2) (r^{\frac{1}{5}} - 2) (r^{\frac{2 \cdot 2}{5}} + r^{\frac{2}{5}} \cdot 4 + 4^{2}) = 0$

Step 5.4.2.2

Multiply $2$ by $2$ .

$(r^{\frac{1}{5}} + 2) (r^{\frac{1}{5}} - 2) (r^{\frac{4}{5}} + r^{\frac{2}{5}} \cdot 4 + 4^{2}) = 0$

Step 5.5

Move $4$ to the left of $r^{\frac{2}{5}}$ .

$(r^{\frac{1}{5}} + 2) (r^{\frac{1}{5}} - 2) (r^{\frac{4}{5}} + 4 r^{\frac{2}{5}} + 4^{2}) = 0$

Step 5.6

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

$(r^{\frac{1}{5}} + 2) (r^{\frac{1}{5}} - 2) (r^{\frac{4}{5}} + 4 r^{\frac{2}{5}} + 16) = 0$

Step 5.7

Reorder terms.

$(r^{\frac{1}{5}} + 2) (r^{\frac{1}{5}} - 2) (4 r^{\frac{2}{5}} + r^{\frac{4}{5}} + 16) = 0$

Step 6

Simplify $(r^{\frac{1}{5}} + 2) (r^{\frac{1}{5}} - 2) (4 r^{\frac{2}{5}} + r^{\frac{4}{5}} + 16)$ .

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Step 6.1

Expand $(r^{\frac{1}{5}} + 2) (r^{\frac{1}{5}} - 2)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

$(r^{\frac{1}{5}} (r^{\frac{1}{5}} - 2) + 2 (r^{\frac{1}{5}} - 2)) (4 r^{\frac{2}{5}} + r^{\frac{4}{5}} + 16) = 0$

Step 6.1.2

Apply the distributive property.

$(r^{\frac{1}{5}} r^{\frac{1}{5}} + r^{\frac{1}{5}} \cdot - 2 + 2 (r^{\frac{1}{5}} - 2)) (4 r^{\frac{2}{5}} + r^{\frac{4}{5}} + 16) = 0$

Step 6.1.3

Apply the distributive property.

$(r^{\frac{1}{5}} r^{\frac{1}{5}} + r^{\frac{1}{5}} \cdot - 2 + 2 r^{\frac{1}{5}} + 2 \cdot - 2) (4 r^{\frac{2}{5}} + r^{\frac{4}{5}} + 16) = 0$

Step 6.2

Simplify terms.

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Step 6.2.1

Combine the opposite terms in $r^{\frac{1}{5}} r^{\frac{1}{5}} + r^{\frac{1}{5}} \cdot - 2 + 2 r^{\frac{1}{5}} + 2 \cdot - 2$ .

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Step 6.2.1.1

Reorder the factors in the terms $r^{\frac{1}{5}} \cdot - 2$ and $2 r^{\frac{1}{5}}$ .

$(r^{\frac{1}{5}} r^{\frac{1}{5}} - 2 r^{\frac{1}{5}} + 2 r^{\frac{1}{5}} + 2 \cdot - 2) (4 r^{\frac{2}{5}} + r^{\frac{4}{5}} + 16) = 0$

Step 6.2.1.2

Add $- 2 r^{\frac{1}{5}}$ and $2 r^{\frac{1}{5}}$ .

$(r^{\frac{1}{5}} r^{\frac{1}{5}} + 0 + 2 \cdot - 2) (4 r^{\frac{2}{5}} + r^{\frac{4}{5}} + 16) = 0$

Step 6.2.1.3

Add $r^{\frac{1}{5}} r^{\frac{1}{5}}$ and $0$ .

$(r^{\frac{1}{5}} r^{\frac{1}{5}} + 2 \cdot - 2) (4 r^{\frac{2}{5}} + r^{\frac{4}{5}} + 16) = 0$

Step 6.2.2

Simplify each term.

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Step 6.2.2.1

Multiply $r^{\frac{1}{5}}$ by $r^{\frac{1}{5}}$ by adding the exponents.

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Step 6.2.2.1.1

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

$(r^{\frac{1}{5} + \frac{1}{5}} + 2 \cdot - 2) (4 r^{\frac{2}{5}} + r^{\frac{4}{5}} + 16) = 0$

Step 6.2.2.1.2

Combine the numerators over the common denominator.

$(r^{\frac{1 + 1}{5}} + 2 \cdot - 2) (4 r^{\frac{2}{5}} + r^{\frac{4}{5}} + 16) = 0$

Step 6.2.2.1.3

Add $1$ and $1$ .

$(r^{\frac{2}{5}} + 2 \cdot - 2) (4 r^{\frac{2}{5}} + r^{\frac{4}{5}} + 16) = 0$

Step 6.2.2.2

Multiply $2$ by $- 2$ .

$(r^{\frac{2}{5}} - 4) (4 r^{\frac{2}{5}} + r^{\frac{4}{5}} + 16) = 0$

Step 6.3

Expand $(r^{\frac{2}{5}} - 4) (4 r^{\frac{2}{5}} + r^{\frac{4}{5}} + 16)$ by multiplying each term in the first expression by each term in the second expression.

$r^{\frac{2}{5}} (4 r^{\frac{2}{5}}) + r^{\frac{2}{5}} r^{\frac{4}{5}} + r^{\frac{2}{5}} \cdot 16 - 4 (4 r^{\frac{2}{5}}) - 4 r^{\frac{4}{5}} - 4 \cdot 16 = 0$

Step 6.4

Simplify terms.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

Rewrite using the commutative property of multiplication.

$4 r^{\frac{2}{5}} r^{\frac{2}{5}} + r^{\frac{2}{5}} r^{\frac{4}{5}} + r^{\frac{2}{5}} \cdot 16 - 4 (4 r^{\frac{2}{5}}) - 4 r^{\frac{4}{5}} - 4 \cdot 16 = 0$

Step 6.4.1.2

Multiply $r^{\frac{2}{5}}$ by $r^{\frac{2}{5}}$ by adding the exponents.

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Step 6.4.1.2.1

Move $r^{\frac{2}{5}}$ .

$4 (r^{\frac{2}{5}} r^{\frac{2}{5}}) + r^{\frac{2}{5}} r^{\frac{4}{5}} + r^{\frac{2}{5}} \cdot 16 - 4 (4 r^{\frac{2}{5}}) - 4 r^{\frac{4}{5}} - 4 \cdot 16 = 0$

Step 6.4.1.2.2

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

$4 r^{\frac{2}{5} + \frac{2}{5}} + r^{\frac{2}{5}} r^{\frac{4}{5}} + r^{\frac{2}{5}} \cdot 16 - 4 (4 r^{\frac{2}{5}}) - 4 r^{\frac{4}{5}} - 4 \cdot 16 = 0$

Step 6.4.1.2.3

Combine the numerators over the common denominator.

$4 r^{\frac{2 + 2}{5}} + r^{\frac{2}{5}} r^{\frac{4}{5}} + r^{\frac{2}{5}} \cdot 16 - 4 (4 r^{\frac{2}{5}}) - 4 r^{\frac{4}{5}} - 4 \cdot 16 = 0$

Step 6.4.1.2.4

Add $2$ and $2$ .

$4 r^{\frac{4}{5}} + r^{\frac{2}{5}} r^{\frac{4}{5}} + r^{\frac{2}{5}} \cdot 16 - 4 (4 r^{\frac{2}{5}}) - 4 r^{\frac{4}{5}} - 4 \cdot 16 = 0$

Step 6.4.1.3

Multiply $r^{\frac{2}{5}}$ by $r^{\frac{4}{5}}$ by adding the exponents.

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Step 6.4.1.3.1

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

$4 r^{\frac{4}{5}} + r^{\frac{2}{5} + \frac{4}{5}} + r^{\frac{2}{5}} \cdot 16 - 4 (4 r^{\frac{2}{5}}) - 4 r^{\frac{4}{5}} - 4 \cdot 16 = 0$

Step 6.4.1.3.2

Combine the numerators over the common denominator.

$4 r^{\frac{4}{5}} + r^{\frac{2 + 4}{5}} + r^{\frac{2}{5}} \cdot 16 - 4 (4 r^{\frac{2}{5}}) - 4 r^{\frac{4}{5}} - 4 \cdot 16 = 0$

Step 6.4.1.3.3

Add $2$ and $4$ .

$4 r^{\frac{4}{5}} + r^{\frac{6}{5}} + r^{\frac{2}{5}} \cdot 16 - 4 (4 r^{\frac{2}{5}}) - 4 r^{\frac{4}{5}} - 4 \cdot 16 = 0$

Step 6.4.1.4

Move $16$ to the left of $r^{\frac{2}{5}}$ .

$4 r^{\frac{4}{5}} + r^{\frac{6}{5}} + 16 \cdot r^{\frac{2}{5}} - 4 (4 r^{\frac{2}{5}}) - 4 r^{\frac{4}{5}} - 4 \cdot 16 = 0$

Step 6.4.1.5

Multiply $4$ by $- 4$ .

$4 r^{\frac{4}{5}} + r^{\frac{6}{5}} + 16 r^{\frac{2}{5}} - 16 r^{\frac{2}{5}} - 4 r^{\frac{4}{5}} - 4 \cdot 16 = 0$

Step 6.4.1.6

Multiply $- 4$ by $16$ .

$4 r^{\frac{4}{5}} + r^{\frac{6}{5}} + 16 r^{\frac{2}{5}} - 16 r^{\frac{2}{5}} - 4 r^{\frac{4}{5}} - 64 = 0$

Step 6.4.2

Combine the opposite terms in $4 r^{\frac{4}{5}} + r^{\frac{6}{5}} + 16 r^{\frac{2}{5}} - 16 r^{\frac{2}{5}} - 4 r^{\frac{4}{5}} - 64$ .

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Step 6.4.2.1

Subtract $16 r^{\frac{2}{5}}$ from $16 r^{\frac{2}{5}}$ .

$4 r^{\frac{4}{5}} + r^{\frac{6}{5}} + 0 - 4 r^{\frac{4}{5}} - 64 = 0$

Step 6.4.2.2

Add $4 r^{\frac{4}{5}} + r^{\frac{6}{5}}$ and $0$ .

$4 r^{\frac{4}{5}} + r^{\frac{6}{5}} - 4 r^{\frac{4}{5}} - 64 = 0$

Step 6.4.2.3

Subtract $4 r^{\frac{4}{5}}$ from $4 r^{\frac{4}{5}}$ .

$r^{\frac{6}{5}} + 0 - 64 = 0$

Step 6.4.2.4

Add $r^{\frac{6}{5}}$ and $0$ .

$r^{\frac{6}{5}} - 64 = 0$

Step 7

Add $64$ to both sides of the equation.

$r^{\frac{6}{5}} = 64$

Step 8

Raise each side of the equation to the power of $\frac{5}{6}$ to eliminate the fractional exponent on the left side.

${(r^{\frac{6}{5}})}^{\frac{5}{6}} = \pm 64^{\frac{5}{6}}$

Step 9

Simplify the exponent.

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Step 9.1

Simplify the left side.

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Step 9.1.1

Simplify ${(r^{\frac{6}{5}})}^{\frac{5}{6}}$ .

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Step 9.1.1.1

Multiply the exponents in ${(r^{\frac{6}{5}})}^{\frac{5}{6}}$ .

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Step 9.1.1.1.1

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

$r^{\frac{6}{5} \cdot \frac{5}{6}} = \pm 64^{\frac{5}{6}}$

Step 9.1.1.1.2

Cancel the common factor of $6$ .

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Step 9.1.1.1.2.1

Cancel the common factor.

$r^{\frac{6}{5} \cdot \frac{5}{6}} = \pm 64^{\frac{5}{6}}$

Step 9.1.1.1.2.2

Rewrite the expression.

$r^{\frac{1}{5} \cdot 5} = \pm 64^{\frac{5}{6}}$

Step 9.1.1.1.3

Cancel the common factor of $5$ .

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Step 9.1.1.1.3.1

Cancel the common factor.

$r^{\frac{1}{5} \cdot 5} = \pm 64^{\frac{5}{6}}$

Step 9.1.1.1.3.2

Rewrite the expression.

$r^{1} = \pm 64^{\frac{5}{6}}$

Step 9.1.1.2

Simplify.

$r = \pm 64^{\frac{5}{6}}$

Step 9.2

Simplify the right side.

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Step 9.2.1

Simplify $\pm 64^{\frac{5}{6}}$ .

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Step 9.2.1.1

Simplify the expression.

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Step 9.2.1.1.1

Rewrite $64$ as $2^{6}$ .

$r = \pm {(2^{6})}^{\frac{5}{6}}$

Step 9.2.1.1.2

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

$r = \pm 2^{6 (\frac{5}{6})}$

Step 9.2.1.2

Cancel the common factor of $6$ .

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Step 9.2.1.2.1

Cancel the common factor.

$r = \pm 2^{6 (\frac{5}{6})}$

Step 9.2.1.2.2

Rewrite the expression.

$r = \pm 2^{5}$

Step 9.2.1.3

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

$r = \pm 32$

Step 10

The complete solution is the result of both the positive and negative portions of the solution.

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Step 10.1

First, use the positive value of the $\pm$ to find the first solution.

$r = 32$

Step 10.2

Next, use the negative value of the $\pm$ to find the second solution.

$r = - 32$

Step 10.3

The complete solution is the result of both the positive and negative portions of the solution.

$r = 32, - 32$