Algebra – Working With Exponents

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.


Table Of Contents


What Is An Exponent?

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.


Exponent Laws

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} and \dfrac{1}{a^{-m}} = a^{m}

Negative Exponents (Version 2): ab^{-m} = \dfrac{a}{b^m} and \dfrac{a}{b^{-m}} = ab^{m}


Notes

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.

When dealing with square roots, remember that \sqrt{x} = x^{1/2}. In general, the n^{th} root \sqrt[n]{x} = x^{1/n}.

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.


Examples

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:

    \[(\dfrac{2}{3})^{-2} = (\dfrac{3}{2})^{2} = \dfrac{3^2}{2^2} = \dfrac{9}{4} \text{ or } 2\dfrac{1}{4}\]


Practice Problems

Here are some practice problems to test your understanding and build your skills. The answers are in the next section.

1) Convert x^{-2}y^{-10} from negative exponents to positive exponents.

2) Evaluate (\dfrac{2}{5})^{-3}.

3) Convert \dfrac{x^{-1}}{z^{-2}} from negative exponents to positive exponents.

4) Convert the fractions \dfrac{1}{x^4} and \dfrac{2}{y^3} into non-fractions with negative exponents.

5) Simplify \dfrac{x^3y^{-2}}{x^7y}.

6) Simplify (\dfrac{x^8}{y^2})^{1/2}.

7) Evaluate (\dfrac{100}{9})^{-1/2}.


Answers

1) x^{-2}y^{-10} = \dfrac{1}{x^{2}y^{10}}

2) (\dfrac{2}{5})^{-3} = (\dfrac{5}{2})^{3} = \dfrac{5^3}{2^3} = \dfrac{125}{8} = 15.625

3) \dfrac{x^{-1}}{z^{-2}} = \dfrac{z^2}{x}

4) \dfrac{1}{x^4} = x^{-4} and \dfrac{2}{y^3} = 2y^{-3}

5) \dfrac{x^3y^{-2}}{x^7y} = x^{3 -7}y^{-2 - 1} = x^{-4}y^{-3} \text{ or } \dfrac{1}{x^4y^3}

6) (\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}

7) (\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}

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