Martinezadam576idn
Answered

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Order the simplification steps of the expression below using the properties of rational exponents.

[tex]\[
\sqrt[3]{875 x^5 y^9}
\][/tex]

[tex]\[
\begin{array}{c}
(875 x^5 y^9)^{\frac{1}{3}} \\
(125 \cdot 7)^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^{\frac{9}{3}} \\
(125)^{\frac{1}{3}} \cdot (7)^{\frac{1}{3}} \cdot x^{\left(\frac{3}{3}+\frac{2}{3}\right)} \cdot y^3 \\
(5^3)^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot x^{\left(1+\frac{2}{3}\right)} \cdot y^3 \\
5 \cdot 7^{\frac{1}{3}} \cdot x \cdot x^{\frac{2}{3}} \cdot y^3 \\
5 x y^3 \cdot \left(7 x^2\right)^{\frac{1}{3}} \\
5 x y^3 \sqrt[3]{7 x^2} \\
\end{array}
\][/tex]

[tex]\[
\begin{array}{c}
\square \\
\downarrow \\
\square \\
\downarrow \\
\square \\
\downarrow \\
\square \\
\end{array}
\][/tex]

Sagot :

GinnyAnsweridn
To systematically simplify the expression [tex]\(\sqrt[3]{875 x^5 y^9}\)[/tex] using the properties of rational exponents, follow these detailed steps:

1. Start with the given expression:
[tex]\[ \left(875 x^5 y^9\right)^{\frac{1}{3}} \][/tex]

2. Rewrite [tex]\(875\)[/tex] as the product of its factors:
[tex]\[ (125 \cdot 7)^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^{\frac{9}{3}} \][/tex]

3. Express [tex]\(125\)[/tex] as [tex]\(5^3\)[/tex]:
[tex]\[ (125)^{\frac{1}{3}} \cdot (7)^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^{\frac{9}{3}} \][/tex]

4. Simplify [tex]\((125)^{\frac{1}{3}}\)[/tex]:
[tex]\[ \left(5^3\right)^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot x^{\left(\frac{3}{3} + \frac{2}{3}\right)} \cdot y^3 \][/tex]

5. Simplify [tex]\(\left(5^3\right)^{\frac{1}{3}}\)[/tex] to get [tex]\(5\)[/tex]:
[tex]\[ 5 \cdot 7^{\frac{1}{3}} \cdot x^{1 + \frac{2}{3}} \cdot y^3 \][/tex]

6. Combine the exponents of [tex]\(x\)[/tex]:
[tex]\[ 5 \cdot 7^{\frac{1}{3}} \cdot x x^{\frac{2}{3}} \cdot y^3 \][/tex]

7. Combine the similar terms:
[tex]\[ 5 x y^3 \cdot \left(7 x^2\right)^{\frac{1}{3}} \][/tex]

8. Rewrite the final simplified expression:
[tex]\[ 5 x y^3 \sqrt[3]{7 x^2} \][/tex]

So the ordered steps in the simplification are:
1. [tex]\(\left(875 x^5 y^9\right)^{\frac{1}{3}}\)[/tex]
2. (125 \cdot 7)^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^{\frac{9}{3}}
3. (125)^{\frac{1}{3}} \cdot (7)^{\frac{1}{3}} \cdot x^{\left(\frac{3}{3} + \frac{2}{3}\right)} \cdot y^3
4. \left(5^3\right)^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot x^{\left(1 + \frac{2}{3}\right)} \cdot y^3
5. 5 \cdot 7^{\frac{1}{3}} \cdot x \cdot x^{\frac{2}{3}} \cdot y^3
6. 5 x y^3 (7 x^2)^{\frac{1}{3}}
7. 5 x y^3 \sqrt[3]{7 x^2}

This sequence orders the simplifications comprehensively, ensuring a step-by-step transformation to the final simplified form.
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