Question

which expression is equivalent to \\(\frac{1}{3}(6x^2y)^2(2x^3y^4)\\) for all values of x and y where the expression is defined?\
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a \\(4x^7y^6\\)\
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b \\(24x^7y^6\\)\
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c \\(4x^{12}y^8\\)\
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d \\(24x^{12}y^8\\)\
\
\\(\circ\\) a\
\\(\circ\\) b\
\\(\circ\\) c\
\\(\circ\\) d

Explanation:

Step1: Expand the squared term

Use exponent rule $(ab)^n=a^nb^n$:
$$(6x^2y)^2 = 6^2 \cdot (x^2)^2 \cdot y^2 = 36x^4y^2$$

Step2: Multiply with the constant term

Multiply by $\frac{1}{3}$:
$$\frac{1}{3} \cdot 36x^4y^2 = 12x^4y^2$$

Step3: Multiply by the remaining term

Multiply by $2x^3y^4$, use $a^m \cdot a^n=a^{m+n}$:
$$12x^4y^2 \cdot 2x^3y^4 = (12 \cdot 2)x^{4+3}y^{2+4} = 24x^7y^6$$

Answer:

B. $24x^7y^6$