High School

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

[tex]\sqrt[3]{875 x^6 y^9}[/tex]

1. [tex](875 x^6 y^9)^{\frac{1}{3}}[/tex]

2. [tex](125 \cdot 7 \cdot x^6 \cdot y^9)^{\frac{1}{3}}[/tex]

3. [tex](125)^{\frac{1}{3}} \cdot (7)^{\frac{1}{3}} \cdot (x^6)^{\frac{1}{3}} \cdot (y^9)^{\frac{1}{3}}[/tex]

4. [tex]5 \cdot 7^{\frac{1}{3}} \cdot x^2 \cdot y^3[/tex]

5. [tex]5x^2y^3 \cdot 7^{\frac{1}{3}}[/tex]

6. [tex]5xy^3 \sqrt[3]{7x^2}[/tex]

7. [tex]5xy^3 \cdot 7^{\frac{1}{3}} \cdot x^{\frac{2}{3}}[/tex]

8. [tex]5xy^3 \cdot (7x^2)^{\frac{2}{3}}[/tex]

9. [tex]5xy^3 \cdot \left(7x^2\right)^{\frac{2}{3}}[/tex]

Answer :

We want to simplify

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

using rational exponents. Here is the step‐by‐step explanation:

------------------------------------------------------------
Step 1. Write the expression with a rational exponent.

We express the cube root as an exponent of [tex]$\frac{1}{3}$[/tex]:

[tex]$$
\sqrt[3]{875\,x^5\,y^9} = \left(875\,x^5\,y^9\right)^{\frac{1}{3}}.
$$[/tex]

------------------------------------------------------------
Step 2. Factor the constant and distribute the exponent.

Notice that the constant [tex]$875$[/tex] factors as

[tex]$$
875 = 5^3 \cdot 7.
$$[/tex]

Then

[tex]$$
\left(875\,x^5\,y^9\right)^{\frac{1}{3}} = \left(5^3\right)^{\frac{1}{3}} \cdot 7^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^{\frac{9}{3}}.
$$[/tex]

Since [tex]$y^{\frac{9}{3}} = y^3$[/tex], and noting that

[tex]$$
x^{\frac{5}{3}} = x^{1+\frac{2}{3}},
$$[/tex]

we rewrite the expression as

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

------------------------------------------------------------
Step 3. Rewrite and group factors.

Rewrite [tex]$x^{1+\frac{2}{3}}$[/tex] as

[tex]$$
x^{1+\frac{2}{3}} = x \cdot x^{\frac{2}{3}}.
$$[/tex]

Then group the factors so that the parts with a cube root are together:

[tex]$$
5 \cdot x \cdot y^3 \cdot \left(7^{\frac{1}{3}} \cdot x^{\frac{2}{3}}\right).
$$[/tex]

------------------------------------------------------------
Step 4. Convert back to radical notation.

Since

[tex]$$
7^{\frac{1}{3}} \cdot x^{\frac{2}{3}} = \sqrt[3]{7\,x^2},
$$[/tex]

the entire expression simplifies to

[tex]$$
5\,x\,y^3\,\sqrt[3]{7\,x^2}.
$$[/tex]

------------------------------------------------------------
Final Answer:

[tex]$$
5x y^3 \sqrt[3]{7 x^2}
$$[/tex]

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