Of course! Let's break down the simplification of the expression [tex]\( \sqrt[4]{567 x^9 y^{11}} \)[/tex] step by step using the properties of exponents.
### Step 1: Rewrite the fourth root as a power of 1/4
The fourth root can be expressed as raising to the power of [tex]\( \frac{1}{4} \)[/tex]:
[tex]\[
\sqrt[4]{567 x^9 y^{11}} = (567 x^9 y^{11})^{\frac{1}{4}}
\][/tex]
### Step 2: Distribute the exponent of [tex]\( \frac{1}{4} \)[/tex] to each factor in the product
We can distribute the exponent of [tex]\( \frac{1}{4} \)[/tex] across the factors inside the parentheses:
[tex]\[
(567 x^9 y^{11})^{\frac{1}{4}} = 567^{\frac{1}{4}} \cdot (x^9)^{\frac{1}{4}} \cdot (y^{11})^{\frac{1}{4}}
\][/tex]
### Step 3: Simplify each term separately
Next, we simplify each term individually:
1. Simplify [tex]\( 567^{\frac{1}{4}} \)[/tex]:
[tex]\[
567^{\frac{1}{4}} \approx 4.87972968509336
\][/tex]
2. Simplify [tex]\( (x^9)^{\frac{1}{4}} \)[/tex] using the property [tex]\( (a^m)^n = a^{mn} \)[/tex]:
[tex]\[
(x^9)^{\frac{1}{4}} = x^{9 \cdot \frac{1}{4}} = x^{\frac{9}{4}}
\][/tex]
3. Simplify [tex]\( (y^{11})^{\frac{1}{4}} \)[/tex] similarly:
[tex]\[
(y^{11})^{\frac{1}{4}} = y^{11 \cdot \frac{1}{4}} = y^{\frac{11}{4}}
\][/tex]
### Step 4: Combine the simplified terms
Finally, we combine the simplified terms to get the fully simplified expression:
[tex]\[
567^{\frac{1}{4}} \cdot (x^9)^{\frac{1}{4}} \cdot (y^{11})^{\frac{1}{4}} = 4.87972968509336 \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}
\][/tex]
Thus, the final simplified form of the expression [tex]\( \sqrt[4]{567 x^9 y^{11}} \)[/tex] is:
[tex]\[
4.87972968509336 \cdot x^{\frac{9}{4}} \cdot y^{\frac{11}{4}}
\][/tex]
I hope this step-by-step explanation helps! If you have any further questions, feel free to ask.
