Solve by Factoring (x-4)^(3/2)=27

${(x - 4)}^{\frac{3}{2}} = 27$

Step 1

Subtract $27$ from both sides of the equation.

${(x - 4)}^{\frac{3}{2}} - 27 = 0$

Step 2

Rewrite ${(x - 4)}^{\frac{3}{2}}$ as ${({(x - 4)}^{\frac{1}{2}})}^{3}$ .

${({(x - 4)}^{\frac{1}{2}})}^{3} - 27 = 0$

Step 3

Rewrite $27$ as $3^{3}$ .

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

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 $a = {(x - 4)}^{\frac{1}{2}}$ and $b = 3$ .

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

Step 5

Simplify.

Multiply the exponents in ${({(x - 4)}^{\frac{1}{2}})}^{2}$ .

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Move $3$ to the left of ${(x - 4)}^{\frac{1}{2}}$ .

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

Raise $3$ to the power of $2$ .

$({(x - 4)}^{\frac{1}{2}} - 3) (x - 4 + 3 {(x - 4)}^{\frac{1}{2}} + 9) = 0$

Add $- 4$ and $9$ .

$({(x - 4)}^{\frac{1}{2}} - 3) (x + 3 {(x - 4)}^{\frac{1}{2}} + 5) = 0$

Step 6

Simplify $({(x - 4)}^{\frac{1}{2}} - 3) (x + 3 {(x - 4)}^{\frac{1}{2}} + 5)$ .

Expand $({(x - 4)}^{\frac{1}{2}} - 3) (x + 3 {(x - 4)}^{\frac{1}{2}} + 5)$ by multiplying each term in the first expression by each term in the second expression.

${(x - 4)}^{\frac{1}{2}} x + {(x - 4)}^{\frac{1}{2}} (3 {(x - 4)}^{\frac{1}{2}}) + {(x - 4)}^{\frac{1}{2}} \cdot 5 - 3 x - 3 (3 {(x - 4)}^{\frac{1}{2}}) - 3 \cdot 5 = 0$

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

${(x - 4)}^{\frac{1}{2}} x + 3 {(x - 4)}^{\frac{1}{2}} {(x - 4)}^{\frac{1}{2}} + {(x - 4)}^{\frac{1}{2}} \cdot 5 - 3 x - 3 (3 {(x - 4)}^{\frac{1}{2}}) - 3 \cdot 5 = 0$

Multiply ${(x - 4)}^{\frac{1}{2}}$ by ${(x - 4)}^{\frac{1}{2}}$ by adding the exponents.

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

${(x - 4)}^{\frac{1}{2}} x + 3 ({(x - 4)}^{\frac{1}{2}} {(x - 4)}^{\frac{1}{2}}) + {(x - 4)}^{\frac{1}{2}} \cdot 5 - 3 x - 3 (3 {(x - 4)}^{\frac{1}{2}}) - 3 \cdot 5 = 0$

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

${(x - 4)}^{\frac{1}{2}} x + 3 {(x - 4)}^{\frac{1}{2} + \frac{1}{2}} + {(x - 4)}^{\frac{1}{2}} \cdot 5 - 3 x - 3 (3 {(x - 4)}^{\frac{1}{2}}) - 3 \cdot 5 = 0$

Combine the numerators over the common denominator.

${(x - 4)}^{\frac{1}{2}} x + 3 {(x - 4)}^{\frac{1 + 1}{2}} + {(x - 4)}^{\frac{1}{2}} \cdot 5 - 3 x - 3 (3 {(x - 4)}^{\frac{1}{2}}) - 3 \cdot 5 = 0$

Add $1$ and $1$ .

${(x - 4)}^{\frac{1}{2}} x + 3 {(x - 4)}^{\frac{2}{2}} + {(x - 4)}^{\frac{1}{2}} \cdot 5 - 3 x - 3 (3 {(x - 4)}^{\frac{1}{2}}) - 3 \cdot 5 = 0$

Divide $2$ by $2$ .

${(x - 4)}^{\frac{1}{2}} x + 3 {(x - 4)}^{1} + {(x - 4)}^{\frac{1}{2}} \cdot 5 - 3 x - 3 (3 {(x - 4)}^{\frac{1}{2}}) - 3 \cdot 5 = 0$

Simplify $3 {(x - 4)}^{1}$ .

${(x - 4)}^{\frac{1}{2}} x + 3 (x - 4) + {(x - 4)}^{\frac{1}{2}} \cdot 5 - 3 x - 3 (3 {(x - 4)}^{\frac{1}{2}}) - 3 \cdot 5 = 0$

Apply the distributive property.

${(x - 4)}^{\frac{1}{2}} x + 3 x + 3 \cdot - 4 + {(x - 4)}^{\frac{1}{2}} \cdot 5 - 3 x - 3 (3 {(x - 4)}^{\frac{1}{2}}) - 3 \cdot 5 = 0$

Multiply $3$ by $- 4$ .

${(x - 4)}^{\frac{1}{2}} x + 3 x - 12 + {(x - 4)}^{\frac{1}{2}} \cdot 5 - 3 x - 3 (3 {(x - 4)}^{\frac{1}{2}}) - 3 \cdot 5 = 0$

Move $5$ to the left of ${(x - 4)}^{\frac{1}{2}}$ .

${(x - 4)}^{\frac{1}{2}} x + 3 x - 12 + 5 \cdot {(x - 4)}^{\frac{1}{2}} - 3 x - 3 (3 {(x - 4)}^{\frac{1}{2}}) - 3 \cdot 5 = 0$

Multiply $3$ by $- 3$ .

${(x - 4)}^{\frac{1}{2}} x + 3 x - 12 + 5 {(x - 4)}^{\frac{1}{2}} - 3 x - 9 {(x - 4)}^{\frac{1}{2}} - 3 \cdot 5 = 0$

Multiply $- 3$ by $5$ .

${(x - 4)}^{\frac{1}{2}} x + 3 x - 12 + 5 {(x - 4)}^{\frac{1}{2}} - 3 x - 9 {(x - 4)}^{\frac{1}{2}} - 15 = 0$

Simplify by adding terms.

Combine the opposite terms in ${(x - 4)}^{\frac{1}{2}} x + 3 x - 12 + 5 {(x - 4)}^{\frac{1}{2}} - 3 x - 9 {(x - 4)}^{\frac{1}{2}} - 15$ .

Subtract $3 x$ from $3 x$ .

${(x - 4)}^{\frac{1}{2}} x - 12 + 5 {(x - 4)}^{\frac{1}{2}} + 0 - 9 {(x - 4)}^{\frac{1}{2}} - 15 = 0$

Add ${(x - 4)}^{\frac{1}{2}} x - 12 + 5 {(x - 4)}^{\frac{1}{2}}$ and $0$ .

${(x - 4)}^{\frac{1}{2}} x - 12 + 5 {(x - 4)}^{\frac{1}{2}} - 9 {(x - 4)}^{\frac{1}{2}} - 15 = 0$

Subtract $15$ from $- 12$ .

${(x - 4)}^{\frac{1}{2}} x - 27 + 5 {(x - 4)}^{\frac{1}{2}} - 9 {(x - 4)}^{\frac{1}{2}} = 0$

Subtract $9 {(x - 4)}^{\frac{1}{2}}$ from $5 {(x - 4)}^{\frac{1}{2}}$ .

${(x - 4)}^{\frac{1}{2}} x - 27 - 4 {(x - 4)}^{\frac{1}{2}} = 0$

Reorder factors in ${(x - 4)}^{\frac{1}{2}} x - 27 - 4 {(x - 4)}^{\frac{1}{2}}$ .

$x {(x - 4)}^{\frac{1}{2}} - 27 - 4 {(x - 4)}^{\frac{1}{2}} = 0$

Step 7

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

$x \approx 13$

Step 8

Solve by Factoring (x-4)^(3/2)=27

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