Solve by Factoring (8x-8)^(3/2)=64

${(8 x - 8)}^{\frac{3}{2}} = 64$

Step 1

Subtract $64$ from both sides of the equation.

${(8 x - 8)}^{\frac{3}{2}} - 64 = 0$

Step 2

Rewrite ${(8 x - 8)}^{\frac{3}{2}}$ as ${({(8 x - 8)}^{\frac{1}{2}})}^{3}$ .

${({(8 x - 8)}^{\frac{1}{2}})}^{3} - 64 = 0$

Step 3

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

${({(8 x - 8)}^{\frac{1}{2}})}^{3} - 4^{3} = 0$

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 $a = {(8 x - 8)}^{\frac{1}{2}}$ and $b = 4$ .

$({(8 x - 8)}^{\frac{1}{2}} - 4) ({({(8 x - 8)}^{\frac{1}{2}})}^{2} + {(8 x - 8)}^{\frac{1}{2}} \cdot 4 + 4^{2}) = 0$

Step 5

Simplify.

Multiply the exponents in ${({(8 x - 8)}^{\frac{1}{2}})}^{2}$ .

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

$({(8 x - 8)}^{\frac{1}{2}} - 4) ({(8 x - 8)}^{\frac{1}{2} \cdot 2} + {(8 x - 8)}^{\frac{1}{2}} \cdot 4 + 4^{2}) = 0$

Cancel the common factor of $2$ .

Cancel the common factor.

$({(8 x - 8)}^{\frac{1}{2}} - 4) ({(8 x - 8)}^{\frac{1}{2} \cdot 2} + {(8 x - 8)}^{\frac{1}{2}} \cdot 4 + 4^{2}) = 0$

Rewrite the expression.

$({(8 x - 8)}^{\frac{1}{2}} - 4) ((8 x - 8) + {(8 x - 8)}^{\frac{1}{2}} \cdot 4 + 4^{2}) = 0$

Simplify.

$({(8 x - 8)}^{\frac{1}{2}} - 4) (8 x - 8 + {(8 x - 8)}^{\frac{1}{2}} \cdot 4 + 4^{2}) = 0$

Move $4$ to the left of ${(8 x - 8)}^{\frac{1}{2}}$ .

$({(8 x - 8)}^{\frac{1}{2}} - 4) (8 x - 8 + 4 {(8 x - 8)}^{\frac{1}{2}} + 4^{2}) = 0$

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

$({(8 x - 8)}^{\frac{1}{2}} - 4) (8 x - 8 + 4 {(8 x - 8)}^{\frac{1}{2}} + 16) = 0$

Add $- 8$ and $16$ .

$({(8 x - 8)}^{\frac{1}{2}} - 4) (8 x + 4 {(8 x - 8)}^{\frac{1}{2}} + 8) = 0$

Factor.

Factor $4$ out of $8 x + 4 {(8 x - 8)}^{\frac{1}{2}} + 8$ .

Factor $4$ out of $8 x$ .

$({(8 x - 8)}^{\frac{1}{2}} - 4) (4 (2 x) + 4 {(8 x - 8)}^{\frac{1}{2}} + 8) = 0$

Factor $4$ out of $4 {(8 x - 8)}^{\frac{1}{2}}$ .

$({(8 x - 8)}^{\frac{1}{2}} - 4) (4 (2 x) + 4 ({(8 x - 8)}^{\frac{1}{2}}) + 8) = 0$

Factor $4$ out of $8$ .

$({(8 x - 8)}^{\frac{1}{2}} - 4) (4 (2 x) + 4 {(8 x - 8)}^{\frac{1}{2}} + 4 \cdot 2) = 0$

Factor $4$ out of $4 (2 x) + 4 {(8 x - 8)}^{\frac{1}{2}}$ .

$({(8 x - 8)}^{\frac{1}{2}} - 4) (4 (2 x + {(8 x - 8)}^{\frac{1}{2}}) + 4 \cdot 2) = 0$

Factor $4$ out of $4 (2 x + {(8 x - 8)}^{\frac{1}{2}}) + 4 \cdot 2$ .

$({(8 x - 8)}^{\frac{1}{2}} - 4) (4 (2 x + {(8 x - 8)}^{\frac{1}{2}} + 2)) = 0$

Remove unnecessary parentheses.

$({(8 x - 8)}^{\frac{1}{2}} - 4) \cdot (4 (2 x + {(8 x - 8)}^{\frac{1}{2}} + 2)) = 0$

Step 6

Simplify $({(8 x - 8)}^{\frac{1}{2}} - 4) \cdot (4 (2 x + {(8 x - 8)}^{\frac{1}{2}} + 2))$ .

Apply the distributive property.

$({(8 x - 8)}^{\frac{1}{2}} - 4) \cdot (4 (2 x) + 4 {(8 x - 8)}^{\frac{1}{2}} + 4 \cdot 2) = 0$

Simplify.

Multiply $2$ by $4$ .

$({(8 x - 8)}^{\frac{1}{2}} - 4) \cdot (8 x + 4 {(8 x - 8)}^{\frac{1}{2}} + 4 \cdot 2) = 0$

Multiply $4$ by $2$ .

$({(8 x - 8)}^{\frac{1}{2}} - 4) \cdot (8 x + 4 {(8 x - 8)}^{\frac{1}{2}} + 8) = 0$

Expand $({(8 x - 8)}^{\frac{1}{2}} - 4) (8 x + 4 {(8 x - 8)}^{\frac{1}{2}} + 8)$ by multiplying each term in the first expression by each term in the second expression.

${(8 x - 8)}^{\frac{1}{2}} (8 x) + {(8 x - 8)}^{\frac{1}{2}} (4 {(8 x - 8)}^{\frac{1}{2}}) + {(8 x - 8)}^{\frac{1}{2}} \cdot 8 - 4 (8 x) - 4 (4 {(8 x - 8)}^{\frac{1}{2}}) - 4 \cdot 8 = 0$

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$8 {(8 x - 8)}^{\frac{1}{2}} x + {(8 x - 8)}^{\frac{1}{2}} (4 {(8 x - 8)}^{\frac{1}{2}}) + {(8 x - 8)}^{\frac{1}{2}} \cdot 8 - 4 (8 x) - 4 (4 {(8 x - 8)}^{\frac{1}{2}}) - 4 \cdot 8 = 0$

Rewrite using the commutative property of multiplication.

$8 {(8 x - 8)}^{\frac{1}{2}} x + 4 {(8 x - 8)}^{\frac{1}{2}} {(8 x - 8)}^{\frac{1}{2}} + {(8 x - 8)}^{\frac{1}{2}} \cdot 8 - 4 (8 x) - 4 (4 {(8 x - 8)}^{\frac{1}{2}}) - 4 \cdot 8 = 0$

Multiply ${(8 x - 8)}^{\frac{1}{2}}$ by ${(8 x - 8)}^{\frac{1}{2}}$ by adding the exponents.

Move ${(8 x - 8)}^{\frac{1}{2}}$ .

$8 {(8 x - 8)}^{\frac{1}{2}} x + 4 ({(8 x - 8)}^{\frac{1}{2}} {(8 x - 8)}^{\frac{1}{2}}) + {(8 x - 8)}^{\frac{1}{2}} \cdot 8 - 4 (8 x) - 4 (4 {(8 x - 8)}^{\frac{1}{2}}) - 4 \cdot 8 = 0$

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

$8 {(8 x - 8)}^{\frac{1}{2}} x + 4 {(8 x - 8)}^{\frac{1}{2} + \frac{1}{2}} + {(8 x - 8)}^{\frac{1}{2}} \cdot 8 - 4 (8 x) - 4 (4 {(8 x - 8)}^{\frac{1}{2}}) - 4 \cdot 8 = 0$

Combine the numerators over the common denominator.

$8 {(8 x - 8)}^{\frac{1}{2}} x + 4 {(8 x - 8)}^{\frac{1 + 1}{2}} + {(8 x - 8)}^{\frac{1}{2}} \cdot 8 - 4 (8 x) - 4 (4 {(8 x - 8)}^{\frac{1}{2}}) - 4 \cdot 8 = 0$

Add $1$ and $1$ .

$8 {(8 x - 8)}^{\frac{1}{2}} x + 4 {(8 x - 8)}^{\frac{2}{2}} + {(8 x - 8)}^{\frac{1}{2}} \cdot 8 - 4 (8 x) - 4 (4 {(8 x - 8)}^{\frac{1}{2}}) - 4 \cdot 8 = 0$

Divide $2$ by $2$ .

$8 {(8 x - 8)}^{\frac{1}{2}} x + 4 {(8 x - 8)}^{1} + {(8 x - 8)}^{\frac{1}{2}} \cdot 8 - 4 (8 x) - 4 (4 {(8 x - 8)}^{\frac{1}{2}}) - 4 \cdot 8 = 0$

Simplify $4 {(8 x - 8)}^{1}$ .

$8 {(8 x - 8)}^{\frac{1}{2}} x + 4 (8 x - 8) + {(8 x - 8)}^{\frac{1}{2}} \cdot 8 - 4 (8 x) - 4 (4 {(8 x - 8)}^{\frac{1}{2}}) - 4 \cdot 8 = 0$

Apply the distributive property.

$8 {(8 x - 8)}^{\frac{1}{2}} x + 4 (8 x) + 4 \cdot - 8 + {(8 x - 8)}^{\frac{1}{2}} \cdot 8 - 4 (8 x) - 4 (4 {(8 x - 8)}^{\frac{1}{2}}) - 4 \cdot 8 = 0$

Multiply $8$ by $4$ .

$8 {(8 x - 8)}^{\frac{1}{2}} x + 32 x + 4 \cdot - 8 + {(8 x - 8)}^{\frac{1}{2}} \cdot 8 - 4 (8 x) - 4 (4 {(8 x - 8)}^{\frac{1}{2}}) - 4 \cdot 8 = 0$

Multiply $4$ by $- 8$ .

$8 {(8 x - 8)}^{\frac{1}{2}} x + 32 x - 32 + {(8 x - 8)}^{\frac{1}{2}} \cdot 8 - 4 (8 x) - 4 (4 {(8 x - 8)}^{\frac{1}{2}}) - 4 \cdot 8 = 0$

Move $8$ to the left of ${(8 x - 8)}^{\frac{1}{2}}$ .

$8 {(8 x - 8)}^{\frac{1}{2}} x + 32 x - 32 + 8 \cdot {(8 x - 8)}^{\frac{1}{2}} - 4 (8 x) - 4 (4 {(8 x - 8)}^{\frac{1}{2}}) - 4 \cdot 8 = 0$

Multiply $8$ by $- 4$ .

$8 {(8 x - 8)}^{\frac{1}{2}} x + 32 x - 32 + 8 {(8 x - 8)}^{\frac{1}{2}} - 32 x - 4 (4 {(8 x - 8)}^{\frac{1}{2}}) - 4 \cdot 8 = 0$

Multiply $4$ by $- 4$ .

$8 {(8 x - 8)}^{\frac{1}{2}} x + 32 x - 32 + 8 {(8 x - 8)}^{\frac{1}{2}} - 32 x - 16 {(8 x - 8)}^{\frac{1}{2}} - 4 \cdot 8 = 0$

Multiply $- 4$ by $8$ .

$8 {(8 x - 8)}^{\frac{1}{2}} x + 32 x - 32 + 8 {(8 x - 8)}^{\frac{1}{2}} - 32 x - 16 {(8 x - 8)}^{\frac{1}{2}} - 32 = 0$

Simplify by adding terms.

Combine the opposite terms in $8 {(8 x - 8)}^{\frac{1}{2}} x + 32 x - 32 + 8 {(8 x - 8)}^{\frac{1}{2}} - 32 x - 16 {(8 x - 8)}^{\frac{1}{2}} - 32$ .

Subtract $32 x$ from $32 x$ .

$8 {(8 x - 8)}^{\frac{1}{2}} x - 32 + 8 {(8 x - 8)}^{\frac{1}{2}} + 0 - 16 {(8 x - 8)}^{\frac{1}{2}} - 32 = 0$

Add $8 {(8 x - 8)}^{\frac{1}{2}} x - 32 + 8 {(8 x - 8)}^{\frac{1}{2}}$ and $0$ .

$8 {(8 x - 8)}^{\frac{1}{2}} x - 32 + 8 {(8 x - 8)}^{\frac{1}{2}} - 16 {(8 x - 8)}^{\frac{1}{2}} - 32 = 0$

Subtract $32$ from $- 32$ .

$8 {(8 x - 8)}^{\frac{1}{2}} x - 64 + 8 {(8 x - 8)}^{\frac{1}{2}} - 16 {(8 x - 8)}^{\frac{1}{2}} = 0$

Subtract $16 {(8 x - 8)}^{\frac{1}{2}}$ from $8 {(8 x - 8)}^{\frac{1}{2}}$ .

$8 {(8 x - 8)}^{\frac{1}{2}} x - 64 - 8 {(8 x - 8)}^{\frac{1}{2}} = 0$

Reorder factors in $8 {(8 x - 8)}^{\frac{1}{2}} x - 64 - 8 {(8 x - 8)}^{\frac{1}{2}}$ .

$8 x {(8 x - 8)}^{\frac{1}{2}} - 64 - 8 {(8 x - 8)}^{\frac{1}{2}} = 0$

Step 7

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 3$

Step 8

Solve by Factoring (8x-8)^(3/2)=64

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