4. Powers, Roots and Radicals

On this page, we’ll continue to revise how numbers work, before applying the procedures to algebra. It all works the same, except that in algebra we use letters to stand for numbers.

Indices

Indices (or powers, or exponents) are very useful in mathematics. Indices are a convenient way of writing multiplications that have many repeated terms.

Example of an Index

For the example 5³, we say that:

5 is the base and

3 is the index (or power, or exponent).

5³ means "multiply 5 by itself 3 times".

[Or more accurately, "multiply 5 by itself repeatedly such that there are three 5’s in the multiplication", or even better, "three 5’s multiplied together". See a discussion on this at Stumbling blocks in math.]

That is, 5³ means

5³ = 5 × 5 × 5

Examples of Integer Exponents

What happens if we have an index of 1, or maybe 0, or even -2?

Let’s set up a pattern using our example above, so we can see what these special cases mean. As we continue this pattern, we are dividing by 5 to get each new line.

5⁴ = 5 × 5 × 5 × 5 = 625

5³ = 5 × 5 × 5 = 125

5² = 5 × 5 = 25

5¹ = 5

5⁰ = 1

5⁻¹ = `1/5`

`5^-2 = 1/5^2 = 1/(5xx5) = 1/25`

`5^-3 = 1/5^3 = 1/(5xx5xx5) = 1/125`

Take note of the special cases

5¹ = 5,

5⁰ = 1, and

`5^-1 = 1/5`

These ones are easy to mess up and they can make you lose sleep unnecessarily when you are doing algebra later.

In general, any number a, (except 0) raised to the power 1 is a.

a¹ = a

Also, any number a, (except 0) raised to the power 0 is 1.

a⁰ = 1

And, any number a, (except 0) raised to the power −1 is `1/a`.

`a^-1=1/a`

Multiplying Numbers With the Same Base

We often need to multiply something like the following:

4³ × 4⁵

We note the numbers have the same base (which is 4) and we think of it as follows:

4³ × 4⁵ ` = \underbrace{(4 xx 4 xx 4)}_{3" of them"} xx \underbrace{(4 xx 4 xx 4 xx 4 xx 4)}_{5" of them"}`

We get 3 fours from the first bracket and 5 fours from the second bracket, so altogether we will have 3 + 5 = 8 fours multiplied together.

4³ × 4⁵ = 4³⁺⁵ = 4⁸ (If anyone cares, the final answer is 65,536. :-)

In general, we can say for any number a and indices m and n:

a^m × aⁿ = a^(m+n)

Dividing Numbers with the Same Base

As an example, let’s divide 3⁶ by 3²:

`{3^6}/{3^2}` `={(3xx3xx3xx3xx3xx3)}/(3xx3)` = 3 × 3 × 3 × 3 = 3⁴ = 81

We cancelled out 2 of the threes on top and the 2 threes on the bottom of the fraction, leaving 4 threes on the top (and the number 1 on the bottom).

In general, for any number a (except 0) and indices m and n:

`{a^m}/a^n=a^(m-n)`

Raising an Index Expression to an Index

As an example, let’s raise the number 4² to the power 3:

(4²)³ = 4² × 4² × 4²

From the multiplication example above, we can see that this is going to give us 4⁶. We could have done this as:

(4²)³ = 4^2×3 = 4⁶

In general, we have for any base a and indices m and n:

(a^m)ⁿ = a^mn

Raising a Product to a Power

Number example:

(5 × 2)³ = 5³ × 2³

In this case, with numbers, it would be better to perform the multiplication in brackets first and then raise our answer to the power 3. But when we are using letters in algebra, we cannot do such a thing and we need to know how to expand it out.

In general:

(ab)ⁿ = aⁿ × bⁿ

Raising a Quotient to a Power

A "quotient" is just an expression involving one number divided by another. In other words, a fraction.

Number example:

`(2/3)^5={2^5}/3^5`

In general:

`(a/b)^n={a^n}/b^n`

Summary of Index Laws

`a^mxxa^n = a^(m+n)`

`a^m/a^n = a^(m-n), (ane0)`

`(a^m)^n = a^(m n)`

`(ab)^n = a^nb^n`

`(a/b)^n = a^n/b^n`

`a^0=1,(ane0)`

`(ab)^n = a^nb^n,(ane0)`

NOTE 1: These rules apply when a and b are positive and m and n are integers. As a counter-example for the third rule `(a^m)^n = a^(m n)`, if `a<0` and `n` is a fraction, we could have:

`[(-3)^2]^(1//2)` means do `(-3)^2 = 9` first, then find square root: `9^(1//2) = 3`.

However,

`(-3)^[2xx(1//2)]` means do `[2xx(1//2)] = 1` first, then raise `(-3)` to the power `1`: `(-3)^1 = -3`.

The final answers are different, so `[(-3)^2]^(1//2) ≠ (-3)^[2xx(1//2)]`.

NOTE 2: We don't have any similar formulas for expressions like `a^m+a^n = ...`

This is because we can only add or subtract like terms (ones that have the same letter part, raised to the same power). For example, this is okay:

`5a^2 + 3a^2 = 8a^2`,

because we are adding like terms.

But we cannot do anything with the following expression:

`5a^3 + 3a^7`

because these are unlike terms (the letter part is raised to a different power). (We can factor this, but cannot expand it in any way or add the terms.)

To see how all this is used in algebra, go to:

• 1. Addition and Subtraction of Algebraic Expressions and
• 2. Multiplication of Algebraic Expressions

Roots and Radicals

We use the radical sign: `sqrt(\ \ )`

It means "square root". The square root is actually a fractional index and is equivalent to raising a number to the power 1/2.

So, for example:

`25^(1/2) = sqrt(25) = 5`

You can also have

Cube root: `root(3)x` (which is equivalent to raising to the power 1/3), and

Fourth root: `root(4)x` (power 1/4) and so on.

See more at Fractional Exponents. See also Simplest Radical Form.

Key things to note:

If a ≥ 0 and b ≥ 0, we have:

`sqrt(axxb)=sqrt(a)xxsqrt(b)`

However, this only works for multiplying. Please note that:

`sqrt(a+b)`

does not equal

`sqrt(a)+sqrt(b)`

(Try it with some real numbers on your calculator).

Also, this one is often found in mathematics:

`sqrt(a^2)=a`

This confuses a lot of students. But it just means:

1. Start with a number
2. Square it
3. Find the square root of the result
4. Finish with the number you started with

For example, start with 3.

Square it, you get 9.

Take the square root, you get 3, which is back where you started.

Why does it matter? Often we need to "undo" a square when solving an equation, so we find the square root of both sides. It’s good to know what you are doing.