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

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

Step 1

Subtract $8$ from both sides of the equation.

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

Step 2

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

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

Step 3

Rewrite $8$ as $2^{3}$ .

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

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

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

Step 5

Simplify.

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

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

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

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

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

Add $- 2$ and $4$ .

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

Step 6

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

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

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

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $1$ and $1$ .

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

Divide $2$ by $2$ .

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

Simplify $2 {(x - 2)}^{1}$ .

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

Apply the distributive property.

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

Multiply $2$ by $- 2$ .

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

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

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

Multiply $2$ by $- 2$ .

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

Multiply $- 2$ by $2$ .

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

Simplify by adding terms.

Combine the opposite terms in ${(x - 2)}^{\frac{1}{2}} x + 2 x - 4 + 2 {(x - 2)}^{\frac{1}{2}} - 2 x - 4 {(x - 2)}^{\frac{1}{2}} - 4$ .

Subtract $2 x$ from $2 x$ .

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

Add ${(x - 2)}^{\frac{1}{2}} x - 4 + 2 {(x - 2)}^{\frac{1}{2}}$ and $0$ .

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

Subtract $4$ from $- 4$ .

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

Subtract $4 {(x - 2)}^{\frac{1}{2}}$ from $2 {(x - 2)}^{\frac{1}{2}}$ .

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

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

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

Step 7

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

$x = 6$

Step 8

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

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