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What is the cube root of [tex][tex]$216 x^9 y^{18}$[/tex][/tex]?

A. [tex]$4 x^3 y^6$[/tex]
B. [tex]$6 x^3 y^6$[/tex]
C. [tex][tex]$72 x^6 y^{15}$[/tex][/tex]
D. [tex]$213 x^6 y^{15}$[/tex]


Sagot :

To determine the cube root of [tex]\( 216 x^9 y^{18} \)[/tex], we need to take the cube root of each factor in the expression separately.

### Step-by-Step Solution:

1. Cube Root of the Constant Term:

The constant term is [tex]\(216\)[/tex]. We need to find the cube root of [tex]\(216\)[/tex]:
[tex]\[ \sqrt[3]{216} = 6 \][/tex]

2. Cube Root of [tex]\( x^9 \)[/tex]:

For the variable [tex]\( x \)[/tex], the original exponent is [tex]\(9\)[/tex]. To find the cube root, we divide this exponent by [tex]\(3\)[/tex]:
[tex]\[ \sqrt[3]{x^9} = x^{\frac{9}{3}} = x^3 \][/tex]

3. Cube Root of [tex]\( y^{18} \)[/tex]:

For the variable [tex]\( y \)[/tex], the original exponent is [tex]\(18\)[/tex]. Similarly, to find the cube root, we divide this exponent by [tex]\(3\)[/tex]:
[tex]\[ \sqrt[3]{y^{18}} = y^{\frac{18}{3}} = y^6 \][/tex]

4. Combining all Results:

Now we combine all the individual cube roots we have calculated:
[tex]\[ \sqrt[3]{216 x^9 y^{18}} = 6 x^3 y^6 \][/tex]

### Conclusion:

The cube root of [tex]\( 216 x^9 y^{18} \)[/tex] is:
[tex]\[ 6 x^3 y^6 \][/tex]

Therefore, the correct choice is:
[tex]\[ \boxed{6 x^3 y^6} \][/tex]
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