Question Solve $\left(z-7\right)^{\frac{4}{3}}=5$ for real number $z$ .

Studdy Solution

STEP 1

Assumptions
1. We are solving the equation $(z-7)^{\frac{4}{3}}=5$ for the real number $z$ .
2. We will use the property that if $a^b = c$ , then $a = c^{\frac{1}{b}}$ .

STEP 2

Isolate the base of the exponent on one side of the equation.
$(z-7)^{\frac{4}{3}} = 5$

STEP 3

Take the cube root of both sides of the equation to eliminate the exponent of $\frac{4}{3}$ . This is equivalent to raising both sides to the power of $\frac{3}{4}$ .
$\left((z-7)^{\frac{4}{3}}\right)^{\frac{3}{4}} = 5^{\frac{3}{4}}$

STEP 4

Simplify the left side of the equation by using the property that $(a^b)^c = a^{bc}$ .
$(z-7)^{\frac{4}{3} \cdot \frac{3}{4}} = (z-7)^1$

STEP 5

Simplify the exponent on the left side of the equation.
$(z-7)^1 = z-7$

STEP 6

Simplify the right side of the equation by calculating $5^{\frac{3}{4}}$ .
$5^{\frac{3}{4}} = \sqrt[4]{5^3}$

STEP 7

Calculate the fourth root of $5^3$ .
$\sqrt[4]{5^3} = \sqrt[4]{125}$

STEP 8

Since we are looking for a real number solution and the fourth root of 125 does not have a simple real number expression, we can leave the right side as $\sqrt[4]{125}$ for now.
$z-7 = \sqrt[4]{125}$

STEP 9

Add 7 to both sides of the equation to solve for $z$ .
$z = \sqrt[4]{125} + 7$

STEP 10

Express the solution for $z$ .
$z = 7 + \sqrt[4]{125}$
Since $\sqrt[4]{125}$ is the real fourth root of 125, this is the solution for $z$ .

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