Power of a Product: Variable in the base of the power

We use the formula:

$(a\times b)^n=a^nb^n$

$(5\times x\times3)^3=(15x)^3$

$(15x)^3=(15\times x)^3$

$15^3x^3$

We use the formula:

$(a\times b)^x=a^xb^x$

Therefore, we obtain:

$(a\times5\times6\times y)^5=(a\times30\times y)^5$

$a^530^5y^5$

We use the formula

$(a\times b)^x=a^xb^x$

Therefore, we obtain:

$a^2b^28^2$

We use the formula:

$(a\times b)^n=a^nb^n$

$(y\times x\times3)^5=y^5x^53^5$

We use the power law for multiplication within parentheses:

$(x\cdot y)^n=x^n\cdot y^n$

We apply it in the problem:

$(y\cdot7\cdot3)^4=y^4\cdot7^4\cdot3^4$

Therefore, the correct answer is option a.

Note:

From the formula of the power property mentioned above, we can understand that it applies not only to two terms within parentheses, but also for multiple terms within parentheses.

Let's solve this in two stages. In the first stage, we'll use the power rule for a power of a product in parentheses:

$(z\cdot t)^n=z^n\cdot t^n$

which states that when raising a product in parentheses to a power, each factor in the product is raised to that power when opening the parentheses,

Let's apply this rule to our problem:

$(x^2\cdot a^3)^{\frac{1}{4}}=(x^2)^{\frac{1}{4}}\cdot(a^3)^{\frac{1}{4}}$

where when opening the parentheses, we applied the power to each factor of the product separately, but since each of these factors is being raised to a power, we did this carefully and used parentheses,

Next, we'll use the power rule for a power of a power:

$(b^m)^n=b^{m\cdot n}$

Let's apply this rule to the expression we got:

$(x^2)^{\frac{1}{4}}\cdot(a^3)^{\frac{1}{4}}=x^{2\cdot\frac{1}{4}}\cdot a^{3\cdot\frac{1}{4}}=x^{\frac{2}{4}}\cdot a^{\frac{3}{4}}=x^{\frac{1}{2}}\cdot a^{\frac{3}{4}}$

where in the second stage we performed the multiplication in the exponents of the factors we obtained, while remembering that multiplying fractions means multiplying their numerators, and then - in the final stage, we simplified the fraction in the power of the first factor in the resulting product.

Therefore, the correct answer is answer A.