Question Solve $(z+4)^{3/2}=3$ for real $z$ . Simplify solution(s).

Studdy Solution

STEP 1

Assumptions
1. We are looking for real solutions to the equation $(z+4)^{\frac{3}{2}}=3$ .
2. The exponent $\frac{3}{2}$ can be interpreted as taking the square root and then cubing the result, or vice versa.
3. We will need to isolate $z$ to solve for it.

STEP 2

To solve the equation, we first need to eliminate the exponent. We can do this by raising both sides of the equation to the power of $\frac{2}{3}$ , which is the reciprocal of $\frac{3}{2}$ .
$(z+4)^{\frac{3}{2} \cdot \frac{2}{3}} = 3^{\frac{2}{3}}$

STEP 3

Simplify the left side of the equation by using the property that $(a^{m})^{n} = a^{m \cdot n}$ , which in this case will give us $(z+4)^1$ .
$(z+4)^1 = 3^{\frac{2}{3}}$

STEP 4

Since raising something to the power of 1 leaves it unchanged, we can simplify the left side to just $z+4$ .
$z+4 = 3^{\frac{2}{3}}$

STEP 5

Now we need to find the value of $3^{\frac{2}{3}}$ . This can be interpreted as the cube root of $3$ squared.
$3^{\frac{2}{3}} = (\sqrt[3]{3})^2$

STEP 6

Calculate the cube root of $3$ , which is just $3^{\frac{1}{3}}$ .
$\sqrt[3]{3} = 3^{\frac{1}{3}}$

STEP 7

Now square the cube root of $3$ to get $3^{\frac{2}{3}}$ .
$(3^{\frac{1}{3}})^2 = (3^{\frac{1}{3}} \cdot 3^{\frac{1}{3}}) = 3^{\frac{1}{3} + \frac{1}{3}} = 3^{\frac{2}{3}}$

STEP 8

Since we cannot simplify $3^{\frac{2}{3}}$ further without approximating, we leave it as is and rewrite the equation.
$z+4 = 3^{\frac{2}{3}}$

STEP 9

To isolate $z$ , we need to subtract $4$ from both sides of the equation.
$z = 3^{\frac{2}{3}} - 4$

STEP 10

Now we have the solution for $z$ .
$z = 3^{\frac{2}{3}} - 4$
Since we are looking for real solutions and $3^{\frac{2}{3}}$ is a real number, the solution is valid.
$z = 3^{\frac{2}{3}} - 4$

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