Question Solve for the value of $x$ in the equation $(x-2)^{\frac{3}{4}}=8$ .

Studdy Solution

STEP 1

1. The equation $(x-2)^{\frac{3}{4}}=8$ can be solved for $x$ by applying inverse operations.
2. Raising both sides of the equation to the power of $\frac{4}{3}$ will eliminate the fractional exponent on the left side.
3. After eliminating the fractional exponent, we can isolate $x$ by performing algebraic operations.

STEP 2

1. Raise both sides of the equation to the power of $\frac{4}{3}$ to eliminate the fractional exponent.
2. Simplify the resulting expression to solve for $x$ .

STEP 3

Raise both sides of the equation to the power of $\frac{4}{3}$ to get rid of the fractional exponent on the left side.
$\left((x-2)^{\frac{3}{4}}\right)^{\frac{4}{3}} = 8^{\frac{4}{3}}$

STEP 4

Simplify the left side using the property that $(a^{m})^{n} = a^{mn}$ .
$(x-2)^{\frac{3}{4} \cdot \frac{4}{3}} = (x-2)^1$

STEP 5

Recognize that $(x-2)^1$ is simply $x-2$ .
$x-2 = 8^{\frac{4}{3}}$

STEP 6

Calculate $8^{\frac{4}{3}}$ by recognizing that $8 = 2^3$ and then applying the power to a power rule.
$8^{\frac{4}{3}} = (2^3)^{\frac{4}{3}}$

STEP 7

Simplify the right side using the property that $(a^{m})^{n} = a^{mn}$ .
$(2^3)^{\frac{4}{3}} = 2^{3 \cdot \frac{4}{3}} = 2^4$

STEP 8

Calculate $2^4$ .
$2^4 = 16$

STEP 9

Substitute $16$ back into the equation.
$x-2 = 16$

STEP 10

Add $2$ to both sides of the equation to isolate $x$ .
$x = 16 + 2$

STEP 11

Simplify the right side to find the value of $x$ .
$x = 18$
The solution to the equation $(x-2)^{\frac{3}{4}}=8$ is $x=18$ .

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