Question

justice found that the fifth root of $243x^{15}y^{5}$ is $3x^{3}y$. is justice correct? explain your reasoning.

choose the correct answer below.

a. justice is not correct because $(3x^{3}y)^{5}=3^{5}(x^{3})^{5}y^{5}=81x^{8}y^{5}$.

b. justice is not correct because $(3x^{3}y)^{5}=3^{5}(x^{3})^{5}y^{5}=81x^{15}y^{5}$.

c. justice is correct because $(3x^{3}y)^{5}=3^{5}(x^{3})^{5}y^{5}=243x^{15}y^{5}$.

d. justice is not correct because $(3x^{3}y)^{5}=3^{5}(x^{3})^{5}y^{5}=243x^{8}y^{5}$.

Explanation:

Step1: Recall exponent rules

To check if \(3x^{3}y\) is the fifth root of \(243x^{15}y^{5}\), we need to raise \(3x^{3}y\) to the fifth power and see if it equals \(243x^{15}y^{5}\). Using the power - of - a - product rule \((ab)^n=a^n b^n\) and the power - of - a - power rule \((a^m)^n=a^{mn}\), we have:

Step2: Calculate \((3x^{3}y)^{5}\)

First, apply the power - of - a - product rule: \((3x^{3}y)^{5}=3^{5}\times(x^{3})^{5}\times y^{5}\)

Then, calculate each part:

• \(3^{5}=243\)
• \((x^{3})^{5}=x^{3\times5}=x^{15}\)
• \(y^{5}\) remains as it is.

So, \(3^{5}\times(x^{3})^{5}\times y^{5}=243\times x^{15}\times y^{5}=243x^{15}y^{5}\)

Answer:

C. Justice is correct because \((3x^{3}y)^{5}=3^{5}(x^{3})^{5}y^{5}=243x^{15}y^{5}\)