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]
[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]
