Recall that exponents are a notation that describes multiplication. For example, \((5)^3\) means multiply by \(5\) three times: \(5\cdot5\cdot5=125\). There are many different rules that this repeated multiplication follows. The following table provides these rules, and the examples that follow show how to apply them.
Rules of Exponents |
|
|---|---|
| Let \(a,b\) be base numbers and \(m,n\) any positive integers. | Product Rule: \(a^m\cdot a^n=a^{m+n}\) | Quotient Rule: \(\displaystyle{\frac{a^m}{a^n}=a^{m-n}}\) |
| Power Rule: \(\left(a^m\right)^n=a^{m\cdot n}\) | Negative Exponent Rule: \(\displaystyle{a^{-m}=\frac{1}{a^m}}\) |
| Distributive Rule (Multiply): \(\left(a\cdot b\right)^m=a^m\cdot b^m\) | Distributive Rule (Divide): \(\displaystyle{\left(\frac{a}{b}\right)^m=\frac{a^m}{b^m}}\) |
| Zero Exponent Rule: \(a^0=1\) | |
Simplfy the expression, the answer should only have positive exponents: \(-3x^3y^3\cdot x^2y\cdot4xy^4\).
Since everything is multiplication, which follows the commutative property, we can organize all the numbers together, then all the like variables together and apply the product rule:
$$\begin{array}{l} -3x^3y^3\cdot x^2y\cdot4xy^4 \\ \implies (-3\cdot4)(x^3\cdot x^2\cdot x)(y^3\cdot y\cdot y^4) \\ \implies (-12)(x^{3+2+1})(y^{3+1+4}) \\ \implies -12x^6y^8\; \color{green}{\checkmark} \end{array}$$Simplfy the expression, the answer should only have positive exponents: \(\displaystyle{\frac{2h^2j^4k^3}{4h^2j^3k^5}}\).
Since everything is multiplication, which follows the commutative property, we can organize all the numbers together, then all the like variables together and apply the quotient rule:
$$\begin{array}{l} \large{\frac{2h^2j^4k^3}{4h^2j^3k^5}} \\ \implies \large{\frac{2}{4}\cdot\frac{h^2}{h^2} \cdot\frac{j^4}{j^3}\cdot\frac{k^3}{k^5}} \\ \implies \frac{1}{2}\cdot h^{2-2}\cdot j^{4-3}\cdot k^{3-5} \\ \implies \frac{1}{2}\cdot h^0\cdot j^1\cdot k^{-2} \end{array}$$Since this produces a zero and negative exponent, application of those rules needs to occur to finish simplifing this expression:
$$\begin{array}{l} \implies \large{\frac{1}{2}\cdot1\cdot j\cdot\frac{1}{k^2}} \\ \implies \large{\frac{j}{2k^2}}\normalsize{\;\color{green}{\checkmark}} \end{array}$$Simplfy the expression, the answer should only have positive exponents: \(\left(4u^2v^{-3}\right)^4\).
To begin, the problem shows multiplication inside the parenthesis by more than one term, so the distributive rule needs to be applied. Then straight application of the power rule, and use of other rules might be needed to further simplify:
$$\begin{array}{l} \left(4u^2v^{-3}\right)^4 \\ (4^4)(u^2)^4(v^{-3})^4 \\ \implies (4\cdot4\cdot4\cdot4)(u^{2\cdot4})(v^{-3\cdot4}) \\ \implies 64(u^8)(v^{-12}) \\ \implies \large{\frac{64u^8}{v^{12}}}\normalsize{\;\color{green}{\checkmark}} \end{array}$$