To solve the expression [tex]\(\frac{60x^{20}y^{24}}{30x^{10}y^{12}}\)[/tex], we need to simplify both the numerical coefficient and the exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] individually.
### Step 1: Simplify the numerical coefficient
The coefficient in the numerator is 60, and the coefficient in the denominator is 30. We simplify this as follows:
[tex]\[
\frac{60}{30} = 2
\][/tex]
So, the simplified coefficient is 2.
### Step 2: Simplify the exponent of [tex]\(x\)[/tex]
The exponent of [tex]\(x\)[/tex] in the numerator is 20, and in the denominator, it is 10. To simplify the exponents of [tex]\(x\)[/tex], we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
20 - 10 = 10
\][/tex]
So, the simplified exponent of [tex]\(x\)[/tex] is [tex]\(10\)[/tex].
### Step 3: Simplify the exponent of [tex]\(y\)[/tex]
The exponent of [tex]\(y\)[/tex] in the numerator is 24, and in the denominator, it is 12. To simplify the exponents of [tex]\(y\)[/tex], we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
24 - 12 = 12
\][/tex]
So, the simplified exponent of [tex]\(y\)[/tex] is [tex]\(12\)[/tex].
### Step 4: Combine the simplified results
Combining the simplified coefficient and exponents, the expression [tex]\(\frac{60x^{20}y^{24}}{30x^{10}y^{12}}\)[/tex] simplifies to:
[tex]\[
2x^{10}y^{12}
\][/tex]
Therefore, the expression equivalent to [tex]\(\frac{60x^{20}y^{24}}{30x^{10}y^{12}}\)[/tex] is:
[tex]\[
2x^{10}y^{12}
\][/tex]
The correct choice is:
[tex]\[
\boxed{2 x^{10} y^{12}}
\][/tex]
