Please show how to solve this with step by step. (4a^2b^–2)^3

We are asked to simplify the expression `(4a^2b^(-2))^3 ` . (Note that we cannot be asked to solve since this is not an equation; there is no equals sign (or inequality, etc...))

We will use the exponent rules. There are a number of different approaches.

(1) First use the product to a power rule: `(xy)^n=x^ny^n ` This is something like a distributive property of exponents over multiplication instead of multiplication over addition.

`(4a^2b^(-2))^3=4^3(a^2)^3(b^(-2))^3 `

** The equals sign here does not create an equation to solve; it indicates an identity or that both sides are equivalent expressions.

(2) Now we will use the power to a power rule: `(x^n)^m=x^(mn) ` This rule comes from the definition of exponents (at least for integral exponents): `(x^2)^3=x^2*x^2*x^2=x^(2*3) `

So `4^3(a^2)^3(b^(-2))^3=64a^6b^(-6) `

(3) Finally we use the negative exponent rule: `a^(-n)=1/(a^n) `

This is often given as `a^(-1)=1/a ` , but we can combine this rule with the power to a power rule. Thus `a^(-n)=(a^n)^(-1)=1/(a^n) `

So we have `64a^6b^(-6)=64a^6*1/b^(6)=(64a^6)/(b^6) `

---------------------------------------------------------------------------

The expression ` (4a^2b^(-2))^3 ` simplifies to the equivalent expression `(64a^6)/b^6 `

---------------------------------------------------------------------------