Solve the following, with step-by-step to show your work: (4x^2 y^3)^4

Sunday, October 2, 2016

Solve the following, with step-by-step to show your work: (4x^2 y^3)^4

We are asked to simplify the expression $\displaystyle{{\left({4}{{x}}^{{2}}{{y}}^{{3}}\right)}}^{{4}}$ . (Note that we are not asked to solve since this is not an equation or inequality; simplifying produces an equivalent expression where solving produces solution(s).)

We will use the exponent rules:

(1) Use the power of a product rule: $\displaystyle{{\left({a}{b}\right)}}^{{n}}={{a}}^{{n}}{{b}}^{{n}}$ .

$\displaystyle{{\left({4}{{x}}^{{2}}{{y}}^{{3}}\right)}}^{{4}}={{4}}^{{4}}{{\left({{x}}^{{2}}\right)}}^{{4}}{{\left({{y}}^{{3}}\right)}}^{{4}}$ ** The equals sign here indicates equivalent expressions, not an equation with solution(s). This will be true for any x,y in the domain.

(2) Now use the power of a power rule: $\displaystyle{{\left({{a}}^{{m}}\right)}}^{{n}}={{a}}^{{{m}\cdot{n}}}$

So we have $\displaystyle{{4}}^{{4}}{{\left({{x}}^{{2}}\right)}}^{{4}}{{\left({{y}}^{{3}}\right)}}^{{4}}={256}{{x}}^{{8}}{{y}}^{{12}}$

This is simplified as there are no negative exponents and no parantheses.

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$\displaystyle{{\left({4}{{x}}^{{2}}{{y}}^{{3}}\right)}}^{{4}}$ is equivalent to (or simplifies as) $\displaystyle{256}{{x}}^{{8}}{{y}}^{{12}}$

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Sunday, October 2, 2016