Question

which expression is equivalent to \\((x^3y^2z)^5\\)?
\\(\bigcirc\\ x^{15}y^{10}z^5\\)
\\(\bigcirc\\ x^8y^7z^5\\)
\\(\bigcirc\\ x^3y^2z^5\\)
\\(\bigcirc\\ x^{15}y^2z\\)

Explanation:

Step1: Apply Power of a Product Rule

The Power of a Product Rule states that \((ab)^n = a^n b^n\). For the expression \((x^{3}y^{2}z)^{5}\), we apply this rule to each factor inside the parentheses. So we get \( (x^{3})^{5}(y^{2})^{5}(z)^{5} \).

Step2: Apply Power of a Power Rule

The Power of a Power Rule states that \((a^m)^n = a^{m\times n}\).

• For \((x^{3})^{5}\), using the rule, we have \(x^{3\times5}=x^{15}\).
• For \((y^{2})^{5}\), using the rule, we have \(y^{2\times5}=y^{10}\).
• For \((z)^{5}\), it remains \(z^{5}\) (since \(z = z^{1}\), so \(z^{1\times5}=z^{5}\)).

Combining these results, we get \(x^{15}y^{10}z^{5}\).

Answer:

A. \(x^{15}y^{10}z^{5}\)