Laws of Exponents

Introduction

Exponentiation is repeated multiplication of a value by itself. The number of repetitions is indicated by the exponent, the little number above the line. So, x² means multiply x by x. And x³ means we multiply 3 x's together. If we need 54 it is 5 × 5 × 5 × 5 = 625.

From these definitions, it becomes clear that the nth power of a constant or variable is found by multiplying the constant or variable by itself n times.

While we're talking exponents, let's tweak our memories and remind ourselves that x means one x to the power 1 so when we multiply x² by x we have ( x · x ) · x which is x³. This illustrates the first law of exponents that says when we multiply the same variable raised to powers, we add the powers. We mustn't however make the common mistake of applying the law to different variables. In other words, x² · y³ doesn't equal xy5 since the "base variables" are not the same.

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Laws of Exponents

 a) x n • x m = x n + m multiply: add exponents b) x n ÷ x m = x n - mdivide: subtract exponents c) (x n ) m = x n mexponentiate: multiply exponents d) x 0 = 1it's x n + x n = 1 e) (x y) n = x n y n f) g) fraction power = root h) i) it's x 0 + x a = 1/x a

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Note: x2 is called x squared, x3 is called x cubed, x n is called x to the n.

Note: the last law (i), about negative exponents, is a source of much confusion.

Remember that negative exponents are liars!

They make us think things are negative when they're not.

A negative exponent creates a fraction.

To eliminate the negative exponent, put 1 over the term.

So,

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Examples with Variables

 a) x 2 · x 4 = (x · x) · (x · x · x · x) = x 6 b) c) (x 2 ) 3 = x2 · x 2 · x2 = x 6 d) e) (xy) 3 = (xy) · (xy) · (xy) = x 3 y³ f) g) h) i)

The next 2 examples show how to work with a mess of layers and exponents,
both positive and negative.

 1) step 1: flip the fraction to get rid of the negative exponent (–3) step 2: cube the fraction; don't forget to cube the constant 3.

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 2) step 1: flip the fraction to get rid of the negative exponent (–2) step 2: move negative exponents terms; cancel 5/10 and z² / z³

Numerical examples

 1) 3 2 · 3 5 = 3 2 + 5 = 3 7. 2) 3) (–3 3 ) 5 = –3 15 4) 1237 0 = 1 5) 6) 7) (3 · 2) 2 = 6 2 = 36 = 3 2 · 2 2 = 9 · 4 = 36 8) 9)

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Addition and Subtraction of Negative Exponents

When dealing strictly with multiplication and division of variables with negative exponents, we can just move things around the fraction line. However, as always with fractions, addition and subtraction require that we find a common denominator.

Example

Find the sum of a – 1 + (3a) – 2 . Rewrite as

Practice

1. Evaluate

a) (16 x 4 ) 1 / 2

c)

### e)

2. Write the following with only positive exponents

a) a – 2 b 3

c)

### d) (x – 5 ) (x 3 y 2 )

e) f) g)

3. Perform the Indicated Operations

 a) b) c)

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Solutions

a)

c)

### e)

 a) b) c) d) e) f) g)

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 a)

 b)

 c)

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