Sure, let's simplify the expression [tex]\(\sqrt[3]{875 x^5 y^9}\)[/tex] step by step using the properties of rational exponents. Here is the detailed breakdown:
1. Starting Expression:
[tex]\[
(\sqrt[3]{875 x^5 y^9})
\][/tex]
2. Express using rational exponents:
[tex]\[
(875 x^5 y^9)^{\frac{1}{3}}
\][/tex]
3. Raise both sides to the power of [tex]\(\frac{2}{3}\)[/tex]:
[tex]\[
(875 x^5 y^9)^{\frac{2}{3}}
\][/tex]
4. Express 875 as a product of two factors:
[tex]\[
875 = 125 \times 7
\][/tex]
So the expression becomes:
[tex]\[
(125 \times 7)^{\frac{2}{3}} \times x^{\frac{5}{3}} \times y^{\frac{9}{3}}
\][/tex]
5. Separate the powers:
[tex]\[
(125)^{\frac{2}{3}} \times (7)^{\frac{2}{3}} \times x^{\frac{5}{3}} \times y^3
\][/tex]
6. Express 125 as a power of 5:
[tex]\[
125 = 5^3
\][/tex]
So the expression becomes:
[tex]\[
(5^3)^{\frac{2}{3}} \times (7)^{\frac{2}{3}} \times x^{\frac{5}{3}} \times y^3
\][/tex]
7. Simplify the exponents:
[tex]\[
5^{2} \times 7^{\frac{2}{3}} \times x^{\frac{5}{3}} \times y^3
\][/tex]
8. Combine the exponents:
[tex]\[
(5^{1+1} \times 7^{\frac{2}{3}} \times x^{\frac{5}{3}} \times y^3)
\][/tex]
9. Combine like terms:
Combine [tex]\( x^{\frac{5}{3}} \)[/tex]:
[tex]\[
5 x y^3 \left(7 x^2\right)^{\frac{1}{3}}
\][/tex]
10. Distribute the exponents:
[tex]\[
5 x y^3 \cdot \left(7^{\frac{1}{3}} x^{\frac{2}{3}}\right)
\][/tex]
11. Simplify to final expression:
[tex]\[
5 x y^3 \left(7 x^2\right)^{\frac{1}{3}}
\][/tex]
Ordering the steps you've provided to represent the correct sequence:
1. [tex]\((875 x^5 y^9)^{\frac{2}{3}}\)[/tex]
2. [tex]\((125 \cdot 7)^{\frac{2}{3}} \cdot x^{\frac{5}{3}} \cdot y^{\frac{9}{3}}\)[/tex]
3. [tex]\((125)^{\frac{2}{3}} \cdot (7)^{\frac{2}{3}} \cdot x^{\frac{5}{3}} \cdot y^3\)[/tex]
4. [tex]\((5^3)^{\frac{2}{3}} \cdot 7^{\frac{2}{3}} \cdot x^{\left(1 + \frac{2}{3}\right)} \cdot y^3\)[/tex]
5. [tex]\(5^1 \cdot 7^{\frac{1}{3}} \cdot x^1 \cdot x^{\frac{2}{3}} \cdot y^3\)[/tex]
6. [tex]\(5 \cdot x \cdot y^3 \cdot \left(7^{\frac{1}{3}} \cdot x^{\frac{2}{3}}\right)\)[/tex]
7. [tex]\(5 x y^3 \cdot \left(7 x^2\right)^{\frac{1}{3}}\)[/tex]
8. [tex]\(5 x y^3 \sqrt[3]{7 x^2}\)[/tex]
This step-by-step sequence maintains the logical progression of simplifying the original expression using properties of rational exponents.
