simplify (2x^-4 y^-2 / 3x^-6 y^-8)^3
To simplify the expression (2x^-4 y^-2 / 3x^-6 y^-8)^3, let's start by applying the exponent rule for an exponent raised to a power.
First, let's simplify the numerator and denominator separately:
Numerator: 2x^-4 y^-2
Denominator: 3x^-6 y^-8
When we have a negative exponent, we can move it to the opposite side of the fraction by changing its sign. So, let's rewrite the numerator and denominator:
Numerator: 2 / (x^4 y^2)
Denominator: 3 / (x^6 y^8)
Now, we can simplify further by raising the fraction to the power of 3. To do this, we raise both the numerator and denominator to the power of 3:
(2 / (x^4 y^2))^3 / (3 / (x^6 y^8))^3
When we raise a fraction to a power, we raise both the numerator and denominator to that power. So, we have:
(2^3 / (x^4 y^2)^3) / (3^3 / (x^6 y^8)^3)
Simplifying further:
(8 / (x^12 y^6)) / (27 / (x^18 y^24))
To divide fractions, we multiply the first fraction by the reciprocal of the second fraction:
(8 / (x^12 y^6)) * ((x^18 y^24) / 27)
Simplifying the numerator:
8 * (x^18 y^24) = 8x^18 y^24
So, the simplified expression is:
(8x^18 y^24) / (27x^12 y^6)