Which of the following is equal to { (x^2y^3)^-2/(x^6y^3z)^2}
To solve this expression, we can simplify the numerator and denominator separately, and then divide the two results.
First, let's simplify the numerator:
(numerator) = (x^2y^3)^-2
Using the exponent rule for exponentiation of a power, we multiply the exponents inside the parentheses by -2:
(numerator) = x^(-2 * 2) * y^(3 * -2)
(numerator) = x^(-4) * y^(-6)
Next, let's simplify the denominator:
(denominator) = (x^6y^3z)^2
Using the exponent rule for exponentiation of a power, we multiply the exponents inside the parentheses by 2:
(denominator) = x^(6 * 2) * y^(3 * 2) * z^2
(denominator) = x^12 * y^6 * z^2
Now we divide the numerator by the denominator:
{ (x^2y^3)^-2 / (x^6y^3z)^2 } = (x^(-4) * y^(-6)) / (x^12 * y^6 * z^2)
Using the exponent rule for division, when dividing exponents with the same base, we subtract the exponents:
= x^(-4 - 12) * y^(-6 - 6) * z^(-2)
= x^-16 * y^-12 * z^-2
Using the exponent rule, when a number or variable has a negative exponent, it can be expressed as its reciprocal with a positive exponent:
= 1/x^16 * 1/y^12 * 1/z^2
Therefore, the simplified expression is 1/(x^16 * y^12 * z^2).