Question Find the expression with rational exponents that equals $\sqrt[4]{81 x^{3} y^{4} z^{8}}$ .

Studdy Solution

STEP 1

Assumptions
1. We need to find the fourth root of the expression $81 x^{3} y^{4} z^{8}$ .
2. We will express the fourth root as a rational exponent, which means that the exponent will be in the form of a fraction.

STEP 2

The fourth root of a number or expression can be expressed as raising that number or expression to the power of $\frac{1}{4}$ . So, we can rewrite the given expression using rational exponents.
$\sqrt[4]{81 x^{3} y^{4} z^{8}} = (81 x^{3} y^{4} z^{8})^{\frac{1}{4}}$

STEP 3

When raising a product to a power, we can apply the power to each factor individually. This means we can apply the exponent $\frac{1}{4}$ to each of the factors $81$ , $x^{3}$ , $y^{4}$ , and $z^{8}$ separately.
$(81 x^{3} y^{4} z^{8})^{\frac{1}{4}} = 81^{\frac{1}{4}} \cdot x^{3 \cdot \frac{1}{4}} \cdot y^{4 \cdot \frac{1}{4}} \cdot z^{8 \cdot \frac{1}{4}}$

STEP 4

Now we simplify each factor separately.
First, we find the fourth root of $81$ .
$81^{\frac{1}{4}} = \sqrt[4]{81}$

STEP 5

Since $81$ is a perfect fourth power ( $3^4 = 81$ ), we can simplify it.
$\sqrt[4]{81} = 3$

STEP 6

Next, we simplify the exponent for $x^{3}$ by multiplying the exponents.
$x^{3 \cdot \frac{1}{4}} = x^{\frac{3}{4}}$

STEP 7

For $y^{4}$ , since the exponent is a multiple of 4, raising it to the power of $\frac{1}{4}$ will cancel out the exponent.
$y^{4 \cdot \frac{1}{4}} = y^{1} = y$

STEP 8

Finally, we simplify the exponent for $z^{8}$ by multiplying the exponents.
$z^{8 \cdot \frac{1}{4}} = z^{2}$

STEP 9

Now we combine all the simplified factors to get the final expression.
$81^{\frac{1}{4}} \cdot x^{\frac{3}{4}} \cdot y \cdot z^{2} = 3 \cdot x^{\frac{3}{4}} \cdot y \cdot z^{2}$

STEP 10

We can now write the final simplified expression with rational exponents.
$\sqrt[4]{81 x^{3} y^{4} z^{8}} = 3 x^{\left(\frac{3}{4}\right)} y z^{2}$
The correct answer is $3 x^{\left(\frac{3}{4}\right)} y z^{2}$ .

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