Hi. This page will be about exponent laws and algebra. This guide is more suited for a high school audience and can be used as a refresher for pre-calculus and calculus students.
Suppose I want to multiply the number 2 by itself 4 times. I can write this as \(2 \times 2 \times 2 \times 2\). If I wanted to multiply 2 by itself 20 times, that would take up a lot of space. Instead of doing \(2 \times 2 \times 2 \times 2 ... \times 2\) with 20 twos and 19 multiplication signs we can expressed this as \(2^{20}\). The 2 is a base while the superscripted number 20 is the exponent.
If I wanted to multiply 5 by itself 4 times I can write \(5 \times 5 \times 5 \times 5\) or simply \(5^{4} = 625\). The 4 here is like a counter of how many times the base number 5 is multiplied by itself. An alternate view would be \(25 \times 25 = 5^{2} \times 5^{2} = 5^{4} = 625\). In this case I can add the exponents 2 and 2 to get 4 as long as the bases are the same.
Mathematics contains a lot of rules (axioms). Some could say that mathematics is like a language. Here are the rules/laws of exponents. (\(m\) and \(n\)) are typically whole numbers)
Multiplying Numbers of The Same Base
\[a^{m} \times a^{n} = a^{m + n}\]
Dividing Numbers of The Same Base
\[\dfrac{a^{m}}{a^{n}} = a^{m - n}\]
Power of A Power (Power Rule)
\[(a^{m})^{n} = a^{mn}\]
Zero Exponent
\[a^{0} = 1\]
because \(\dfrac{a^{m}}{a^{m}} = a^{m - m} = a^{0} = 1\)
Negative Exponents
\[a^{-m} = \dfrac{1}{a^m} \text{ and }\dfrac{1}{a^{-m}} = a^{m}\]
Negative Exponents (Version 2)
\[ab^{-m} = \dfrac{a}{b^m} \text{ and } \dfrac{a}{b^{-m}} = ab^{m}\]
There are times when you may have to apply multiple exponent laws. For example we could have \((\dfrac{a}{b^{2}})^{-2} = (\dfrac{b^2}{a})^{2} = \dfrac{(b^2)^{2}}{a^2} = \dfrac{b^4}{a^2}\). This example applies the negative exponent then the power rule.
It is important to note that that \((ab)^2 = a^{2}b^{2}\) which is different from \(ab^2\). The exponent 2 is applied to \(ab\) inside the bracket in \((ab)^2\) while the exponent 2 is applied to only \(b\) in \(ab^2\).
In general, the n\(^{th}\) root \(\sqrt[n]{x} = x^{1/n}\). With a square root you would have \(n = 2\).
Recall that \(\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \times \dfrac{d}{c}\). A special case of this would be \(\dfrac{1}{1} \div \dfrac{1}{a} = 1 \div \dfrac{1}{a} = 1 \times \dfrac{a}{1} = 1 \times a = a\).
Here are some examples using various exponent laws.
Example One
Convert \(\dfrac{x^{-2}y^2}{z^{-1}}\) from negative exponents to positive exponents.
Answer:
\[\dfrac{x^{-2}y^2}{z^{-1}} = y^2 \times \dfrac{1}{\dfrac{1}{z}} \times \dfrac{1}{x^2} = y^2 \times z \times \dfrac{1}{x^2} = \dfrac{y^2z}{x^2} \]
Example Two
Simplify \(\dfrac{x^2}{y^{-5}} \times \dfrac{x^3}{y^2}\).
Answer:
\[\dfrac{x^2}{y^{-5}} \times \dfrac{x^3}{y^2} = x^{2 + 3} \times y^5 \times \dfrac{1}{y^2} = x^{5} \times \dfrac{y^5}{y^2} = x^5y^{5 - 2} = x^{5}y^{3}\]
Example Three
Evaluate \((\dfrac{2}{3})^{-2}\).
Answer:
\[(\dfrac{2}{3})^{-2} = (\dfrac{3}{2})^{2} = \dfrac{3^2}{2^2} = \dfrac{9}{4} \text{ or } 2\dfrac{1}{4}\]
Here are some practice problems to test your understanding and build your skills. The answers are in the next section.
Convert \(x^{-2}y^{-10}\) from negative exponents to positive exponents.
Evaluate \((\dfrac{2}{5})^{-3}\).
Convert \(\dfrac{x^{-1}}{z^{-2}}\) from negative exponents to positive exponents.
Convert the fractions \(\dfrac{1}{x^4}\) and \(\dfrac{2}{y^3}\) into non-fractions with negative exponents.
Simplify \(\dfrac{x^3y^{-2}}{x^7y}\).
Simplify \((\dfrac{x^8}{y^2})^{1/2}\).
Evaluate \((\dfrac{100}{9})^{-1/2}\).
\(x^{-2}y^{-10} = \dfrac{1}{x^{2}y^{10}}\)
\((\dfrac{2}{5})^{-3} = (\dfrac{5}{2})^{3} = \dfrac{5^3}{2^3} = \dfrac{125}{8} = 15.625\)
\(\dfrac{x^{-1}}{z^{-2}} = \dfrac{z^2}{x}\)
\(\dfrac{1}{x^4} = x^{-4}\) and \(\dfrac{2}{y^3} = 2y^{-3}\)
\(\dfrac{x^3y^{-2}}{x^7y} = x^{3 -7}y^{-2 - 1} = x^{-4}y^{-3} \text{ or } \dfrac{1}{x^4y^3}\)
\((\dfrac{x^8}{y^2})^{1/2} = \dfrac{(x^{8})^{1/2}}{(y^2)^{1/2}} = \dfrac{x^{8/2}}{y^{2/2}} = \dfrac{x^4}{y}\)
\((\dfrac{100}{9})^{-1/2} = (\dfrac{9}{100})^{1/2} = \dfrac{9^{1/2}}{100^{1/2}} = \dfrac{\sqrt{9}}{\sqrt{100}} = \dfrac{3}{10}\)