Solve by Factoring r^(6/7)=64

$r^{\frac{6}{7}} = 64$

Move $64$ to the left side of the equation by subtracting it from both sides.

$r^{\frac{6}{7}} - 64 = 0$

Rewrite $r^{\frac{6}{7}}$ as ${(r^{\frac{2}{7}})}^{3}$ .

${(r^{\frac{2}{7}})}^{3} - 64 = 0$

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

${(r^{\frac{2}{7}})}^{3} - 4^{3} = 0$

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 $a = r^{\frac{2}{7}}$ and $b = 4$ .

$(r^{\frac{2}{7}} - 4) ({(r^{\frac{2}{7}})}^{2} + r^{\frac{2}{7}} \cdot 4 + 4^{2}) = 0$

Simplify.

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Rewrite $r^{\frac{2}{7}}$ as ${(r^{\frac{1}{7}})}^{2}$ .

$({(r^{\frac{1}{7}})}^{2} - 4) ({(r^{\frac{2}{7}})}^{2} + r^{\frac{2}{7}} \cdot 4 + 4^{2}) = 0$

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

$({(r^{\frac{1}{7}})}^{2} - 2^{2}) ({(r^{\frac{2}{7}})}^{2} + r^{\frac{2}{7}} \cdot 4 + 4^{2}) = 0$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = r^{\frac{1}{7}}$ and $b = 2$ .

$(r^{\frac{1}{7}} + 2) (r^{\frac{1}{7}} - 2) ({(r^{\frac{2}{7}})}^{2} + r^{\frac{2}{7}} \cdot 4 + 4^{2}) = 0$

Multiply the exponents in ${(r^{\frac{2}{7}})}^{2}$ .

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

$(r^{\frac{1}{7}} + 2) (r^{\frac{1}{7}} - 2) (r^{\frac{2}{7} \cdot 2} + r^{\frac{2}{7}} \cdot 4 + 4^{2}) = 0$

Multiply $\frac{2}{7} \cdot 2$ .

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Combine $\frac{2}{7}$ and $2$ .

$(r^{\frac{1}{7}} + 2) (r^{\frac{1}{7}} - 2) (r^{\frac{2 \cdot 2}{7}} + r^{\frac{2}{7}} \cdot 4 + 4^{2}) = 0$

Multiply $2$ by $2$ .

$(r^{\frac{1}{7}} + 2) (r^{\frac{1}{7}} - 2) (r^{\frac{4}{7}} + r^{\frac{2}{7}} \cdot 4 + 4^{2}) = 0$

Move $4$ to the left of $r^{\frac{2}{7}}$ .

$(r^{\frac{1}{7}} + 2) (r^{\frac{1}{7}} - 2) (r^{\frac{4}{7}} + 4 r^{\frac{2}{7}} + 4^{2}) = 0$

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

$(r^{\frac{1}{7}} + 2) (r^{\frac{1}{7}} - 2) (r^{\frac{4}{7}} + 4 r^{\frac{2}{7}} + 16) = 0$

Reorder terms.

$(r^{\frac{1}{7}} + 2) (r^{\frac{1}{7}} - 2) (4 r^{\frac{2}{7}} + r^{\frac{4}{7}} + 16) = 0$

Simplify $(r^{\frac{1}{7}} + 2) (r^{\frac{1}{7}} - 2) (4 r^{\frac{2}{7}} + r^{\frac{4}{7}} + 16)$ .

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Expand $(r^{\frac{1}{7}} + 2) (r^{\frac{1}{7}} - 2)$ using the FOIL Method.

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Apply the distributive property.

$(r^{\frac{1}{7}} (r^{\frac{1}{7}} - 2) + 2 (r^{\frac{1}{7}} - 2)) (4 r^{\frac{2}{7}} + r^{\frac{4}{7}} + 16) = 0$

Apply the distributive property.

$(r^{\frac{1}{7}} r^{\frac{1}{7}} + r^{\frac{1}{7}} \cdot - 2 + 2 (r^{\frac{1}{7}} - 2)) (4 r^{\frac{2}{7}} + r^{\frac{4}{7}} + 16) = 0$

Apply the distributive property.

$(r^{\frac{1}{7}} r^{\frac{1}{7}} + r^{\frac{1}{7}} \cdot - 2 + 2 r^{\frac{1}{7}} + 2 \cdot - 2) (4 r^{\frac{2}{7}} + r^{\frac{4}{7}} + 16) = 0$

Simplify terms.

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Combine the opposite terms in $r^{\frac{1}{7}} r^{\frac{1}{7}} + r^{\frac{1}{7}} \cdot - 2 + 2 r^{\frac{1}{7}} + 2 \cdot - 2$ .

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Reorder the factors in the terms $r^{\frac{1}{7}} \cdot - 2$ and $2 r^{\frac{1}{7}}$ .

$(r^{\frac{1}{7}} r^{\frac{1}{7}} - 2 r^{\frac{1}{7}} + 2 r^{\frac{1}{7}} + 2 \cdot - 2) (4 r^{\frac{2}{7}} + r^{\frac{4}{7}} + 16) = 0$

Add $- 2 r^{\frac{1}{7}}$ and $2 r^{\frac{1}{7}}$ .

$(r^{\frac{1}{7}} r^{\frac{1}{7}} + 0 + 2 \cdot - 2) (4 r^{\frac{2}{7}} + r^{\frac{4}{7}} + 16) = 0$

Add $r^{\frac{1}{7}} r^{\frac{1}{7}}$ and $0$ .

$(r^{\frac{1}{7}} r^{\frac{1}{7}} + 2 \cdot - 2) (4 r^{\frac{2}{7}} + r^{\frac{4}{7}} + 16) = 0$

Simplify each term.

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Multiply $r^{\frac{1}{7}}$ by $r^{\frac{1}{7}}$ by adding the exponents.

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

$(r^{\frac{1}{7} + \frac{1}{7}} + 2 \cdot - 2) (4 r^{\frac{2}{7}} + r^{\frac{4}{7}} + 16) = 0$

Combine the numerators over the common denominator.

$(r^{\frac{1 + 1}{7}} + 2 \cdot - 2) (4 r^{\frac{2}{7}} + r^{\frac{4}{7}} + 16) = 0$

Add $1$ and $1$ .

$(r^{\frac{2}{7}} + 2 \cdot - 2) (4 r^{\frac{2}{7}} + r^{\frac{4}{7}} + 16) = 0$

Multiply $2$ by $- 2$ .

$(r^{\frac{2}{7}} - 4) (4 r^{\frac{2}{7}} + r^{\frac{4}{7}} + 16) = 0$

Expand $(r^{\frac{2}{7}} - 4) (4 r^{\frac{2}{7}} + r^{\frac{4}{7}} + 16)$ by multiplying each term in the first expression by each term in the second expression.

$r^{\frac{2}{7}} (4 r^{\frac{2}{7}}) + r^{\frac{2}{7}} r^{\frac{4}{7}} + r^{\frac{2}{7}} \cdot 16 - 4 (4 r^{\frac{2}{7}}) - 4 r^{\frac{4}{7}} - 4 \cdot 16 = 0$

Simplify terms.

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Simplify each term.

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Rewrite using the commutative property of multiplication.

$4 r^{\frac{2}{7}} r^{\frac{2}{7}} + r^{\frac{2}{7}} r^{\frac{4}{7}} + r^{\frac{2}{7}} \cdot 16 - 4 (4 r^{\frac{2}{7}}) - 4 r^{\frac{4}{7}} - 4 \cdot 16 = 0$

Multiply $r^{\frac{2}{7}}$ by $r^{\frac{2}{7}}$ by adding the exponents.

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Move $r^{\frac{2}{7}}$ .

$4 (r^{\frac{2}{7}} r^{\frac{2}{7}}) + r^{\frac{2}{7}} r^{\frac{4}{7}} + r^{\frac{2}{7}} \cdot 16 - 4 (4 r^{\frac{2}{7}}) - 4 r^{\frac{4}{7}} - 4 \cdot 16 = 0$

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

$4 r^{\frac{2}{7} + \frac{2}{7}} + r^{\frac{2}{7}} r^{\frac{4}{7}} + r^{\frac{2}{7}} \cdot 16 - 4 (4 r^{\frac{2}{7}}) - 4 r^{\frac{4}{7}} - 4 \cdot 16 = 0$

Combine the numerators over the common denominator.

$4 r^{\frac{2 + 2}{7}} + r^{\frac{2}{7}} r^{\frac{4}{7}} + r^{\frac{2}{7}} \cdot 16 - 4 (4 r^{\frac{2}{7}}) - 4 r^{\frac{4}{7}} - 4 \cdot 16 = 0$

Add $2$ and $2$ .

$4 r^{\frac{4}{7}} + r^{\frac{2}{7}} r^{\frac{4}{7}} + r^{\frac{2}{7}} \cdot 16 - 4 (4 r^{\frac{2}{7}}) - 4 r^{\frac{4}{7}} - 4 \cdot 16 = 0$

Multiply $r^{\frac{2}{7}}$ by $r^{\frac{4}{7}}$ by adding the exponents.

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

$4 r^{\frac{4}{7}} + r^{\frac{2}{7} + \frac{4}{7}} + r^{\frac{2}{7}} \cdot 16 - 4 (4 r^{\frac{2}{7}}) - 4 r^{\frac{4}{7}} - 4 \cdot 16 = 0$

Combine the numerators over the common denominator.

$4 r^{\frac{4}{7}} + r^{\frac{2 + 4}{7}} + r^{\frac{2}{7}} \cdot 16 - 4 (4 r^{\frac{2}{7}}) - 4 r^{\frac{4}{7}} - 4 \cdot 16 = 0$

Add $2$ and $4$ .

$4 r^{\frac{4}{7}} + r^{\frac{6}{7}} + r^{\frac{2}{7}} \cdot 16 - 4 (4 r^{\frac{2}{7}}) - 4 r^{\frac{4}{7}} - 4 \cdot 16 = 0$

Move $16$ to the left of $r^{\frac{2}{7}}$ .

$4 r^{\frac{4}{7}} + r^{\frac{6}{7}} + 16 \cdot r^{\frac{2}{7}} - 4 (4 r^{\frac{2}{7}}) - 4 r^{\frac{4}{7}} - 4 \cdot 16 = 0$

Multiply $4$ by $- 4$ .

$4 r^{\frac{4}{7}} + r^{\frac{6}{7}} + 16 r^{\frac{2}{7}} - 16 r^{\frac{2}{7}} - 4 r^{\frac{4}{7}} - 4 \cdot 16 = 0$

Multiply $- 4$ by $16$ .

$4 r^{\frac{4}{7}} + r^{\frac{6}{7}} + 16 r^{\frac{2}{7}} - 16 r^{\frac{2}{7}} - 4 r^{\frac{4}{7}} - 64 = 0$

Combine the opposite terms in $4 r^{\frac{4}{7}} + r^{\frac{6}{7}} + 16 r^{\frac{2}{7}} - 16 r^{\frac{2}{7}} - 4 r^{\frac{4}{7}} - 64$ .

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Subtract $16 r^{\frac{2}{7}}$ from $16 r^{\frac{2}{7}}$ .

$4 r^{\frac{4}{7}} + r^{\frac{6}{7}} + 0 - 4 r^{\frac{4}{7}} - 64 = 0$

Add $4 r^{\frac{4}{7}} + r^{\frac{6}{7}}$ and $0$ .

$4 r^{\frac{4}{7}} + r^{\frac{6}{7}} - 4 r^{\frac{4}{7}} - 64 = 0$

Subtract $4 r^{\frac{4}{7}}$ from $4 r^{\frac{4}{7}}$ .

$r^{\frac{6}{7}} + 0 - 64 = 0$

Add $r^{\frac{6}{7}}$ and $0$ .

$r^{\frac{6}{7}} - 64 = 0$

Add $64$ to both sides of the equation.

$r^{\frac{6}{7}} = 64$

Raise each side of the equation to the $\frac{7}{6}$ power to eliminate the fractional exponent on the left side.

${(r^{\frac{6}{7}})}^{\frac{7}{6}} = \pm {(64)}^{\frac{7}{6}}$

Solve the equation for $r$ .

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Simplify the right side of the equation.

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Simplify the expression.

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

$r = \pm {(2^{6})}^{\frac{7}{6}}$

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

$r = \pm 2^{6 (\frac{7}{6})}$

Cancel the common factor of $6$ .

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Cancel the common factor.

$r = \pm 2^{6 (\frac{7}{6})}$

Rewrite the expression.

$r = \pm 2^{7}$

Raise $2$ to the power of $7$ .

$r = \pm 128$

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

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

$r = 128$

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

$r = - 128$

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

$r = 128, - 128$

Solve by Factoring r^(6/7)=64

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