Answer :
We need to order the simplification steps for the given expression [tex]\( \sqrt[3]{875 x^5 y^9} \)[/tex] using the properties of rational exponents. Below is the step-by-step simplification:
1. Express the radical as a rational exponent:
[tex]\[ \sqrt[3]{875 x^5 y^9} = \left(875 x^5 y^9\right)^{\frac{1}{3}} \][/tex]
2. Break down the factors of 875:
[tex]\[ 875 = 5^3 \times 7 \][/tex]
Thus:
[tex]\[ \left(875 x^5 y^9\right)^{\frac{1}{3}} = \left(5^3 \cdot 7 \cdot x^5 \cdot y^9\right)^{\frac{1}{3}} \][/tex]
3. Apply the rational exponent to each component:
[tex]\[ \left(5^3 \cdot 7 \cdot x^5 \cdot y^9\right)^{\frac{1}{3}} = \left(5^3\right)^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot \left(x^5\right)^{\frac{1}{3}} \cdot \left(y^9\right)^{\frac{1}{3}} \][/tex]
4. Simplify each component:
[tex]\[ \left(5^3\right)^{\frac{1}{3}} = 5 \][/tex]
[tex]\[ \left(x^5\right)^{\frac{1}{3}} = x^{\frac{5}{3}} \][/tex]
[tex]\[ \left(y^9\right)^{\frac{1}{3}} = y^3 \][/tex]
So:
[tex]\[ 5 \cdot 7^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^3 \][/tex]
5. Combine the terms:
[tex]\[ 5 \cdot 7^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^3 \][/tex]
6. Separate and combine like terms further:
[tex]\[ 5 \cdot 7^{\frac{1}{3}} \cdot x^{1 + \frac{2}{3}} \cdot y^3 \][/tex]
Since [tex]\( x^{\frac{5}{3}} = x^{1 + \frac{2}{3}} \)[/tex]:
7. Final form of the expression:
[tex]\[ = 5 x y^3 (7 x^2)^{\frac{1}{3}} \][/tex]
This can also be written as:
[tex]\[ 5 x y^3 \sqrt[3]{7 x^2} \][/tex]
Thus, the ordered steps of the simplification are:
1. [tex]\(\left(875 x^5 y^9\right)^{\frac{1}{3}}\)[/tex]
2. [tex]\((125 \cdot 7)^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^{\frac{9}{3}}\)[/tex]
3. [tex]\((125)^{\frac{1}{3}} \cdot(7)^{\frac{1}{3}} \cdot x^{\left(\frac{3}{3}+\frac{2}{3}\right)} \cdot y^3\)[/tex]
4. [tex]\(\left(5^3\right)^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot x^{\left(1+\frac{2}{3}\right)} \cdot y^3\)[/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]
1. Express the radical as a rational exponent:
[tex]\[ \sqrt[3]{875 x^5 y^9} = \left(875 x^5 y^9\right)^{\frac{1}{3}} \][/tex]
2. Break down the factors of 875:
[tex]\[ 875 = 5^3 \times 7 \][/tex]
Thus:
[tex]\[ \left(875 x^5 y^9\right)^{\frac{1}{3}} = \left(5^3 \cdot 7 \cdot x^5 \cdot y^9\right)^{\frac{1}{3}} \][/tex]
3. Apply the rational exponent to each component:
[tex]\[ \left(5^3 \cdot 7 \cdot x^5 \cdot y^9\right)^{\frac{1}{3}} = \left(5^3\right)^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot \left(x^5\right)^{\frac{1}{3}} \cdot \left(y^9\right)^{\frac{1}{3}} \][/tex]
4. Simplify each component:
[tex]\[ \left(5^3\right)^{\frac{1}{3}} = 5 \][/tex]
[tex]\[ \left(x^5\right)^{\frac{1}{3}} = x^{\frac{5}{3}} \][/tex]
[tex]\[ \left(y^9\right)^{\frac{1}{3}} = y^3 \][/tex]
So:
[tex]\[ 5 \cdot 7^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^3 \][/tex]
5. Combine the terms:
[tex]\[ 5 \cdot 7^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^3 \][/tex]
6. Separate and combine like terms further:
[tex]\[ 5 \cdot 7^{\frac{1}{3}} \cdot x^{1 + \frac{2}{3}} \cdot y^3 \][/tex]
Since [tex]\( x^{\frac{5}{3}} = x^{1 + \frac{2}{3}} \)[/tex]:
7. Final form of the expression:
[tex]\[ = 5 x y^3 (7 x^2)^{\frac{1}{3}} \][/tex]
This can also be written as:
[tex]\[ 5 x y^3 \sqrt[3]{7 x^2} \][/tex]
Thus, the ordered steps of the simplification are:
1. [tex]\(\left(875 x^5 y^9\right)^{\frac{1}{3}}\)[/tex]
2. [tex]\((125 \cdot 7)^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^{\frac{9}{3}}\)[/tex]
3. [tex]\((125)^{\frac{1}{3}} \cdot(7)^{\frac{1}{3}} \cdot x^{\left(\frac{3}{3}+\frac{2}{3}\right)} \cdot y^3\)[/tex]
4. [tex]\(\left(5^3\right)^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot x^{\left(1+\frac{2}{3}\right)} \cdot y^3\)[/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]
