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Select the correct answer.

Use the power of a product property to answer the question. Which expression equals [tex]$(7 y)^{\frac{1}{3}}$[/tex]?

A. [tex]7 y^{\frac{1}{3}}[/tex]
B. [tex]7 y^{\frac{2}{3}}[/tex]
C. [tex]\frac{1}{7^3 y^3}[/tex]
D. [tex]7^{\frac{1}{3}} y^{\frac{1}{3}}[/tex]


Sagot :

To solve the problem using the power of a product property, let's recall how this property works. The property states:

[tex]\[ (ab)^n = a^n \cdot b^n \][/tex]

In other words, when you have a product inside a power, you can apply the exponent to each factor separately.

Given the expression [tex]\((7y)^{1/3}\)[/tex], we apply the power of a product property as follows:

[tex]\[ (7y)^{1/3} = 7^{1/3} \cdot y^{1/3} \][/tex]

Therefore, [tex]\((7y)^{1/3}\)[/tex] simplifies to [tex]\(7^{1/3} \cdot y^{1/3}\)[/tex].

Now, let's match this with the given answer options:

1. [tex]\(7 y^{\frac{1}{3}}\)[/tex] - This expression does not correctly apply the exponent [tex]\( \frac{1}{3} \)[/tex] to both 7 and [tex]\( y \)[/tex].
2. [tex]\(7 y^{\frac{2}{3}}\)[/tex] - This expression applies the wrong exponent to [tex]\( y \)[/tex] and does not apply the exponent to 7.
3. [tex]\(\frac{1}{7^3 y^3}\)[/tex] - This expression suggests taking the reciprocal after cubing 7 and [tex]\( y \)[/tex], which is incorrect.
4. [tex]\(7^{\frac{1}{3}} y^{\frac{1}{3}}\)[/tex] - This is the correct simplification according to the power of a product property.

Thus, the correct answer is:

[tex]\[ \boxed{7^{\frac{1}{3}} y^{\frac{1}{3}}} \][/tex]