What Do Powers/Indices Mean?

Indices (Powers) & Roots

What do Powers/Indices mean?:

is ‘to the power of 2‘ and means (we usually say ‘squared’)

is ‘to the power of 3’ and means (we usually say ‘cubed)

is ‘ to the power of 4’ and means

etc….

The superscript numbers (2, 3 & 4 above) are known as indices or powers. When the power is 2 we say “squared”, when the power is 3 we say “cubed” and for all other powers we say “to the power of….”

Examples:

(3 squared)

(5 cubed)

(3 to the power of 4)

(2 to the power of 5)

(2 to the power of 1)

What do roots mean?:

Roots are the opposite of powers.

(square root of 9) has 2 answers

We know that , so reversing it gives: = 3

Also notice: = 9 so also =

This is often written as (meaning +3 or -3)

(cubed root of 27) = 3

We know: 3 x 3 x 3 = 33 = 27, therefore = 3

Also notice: so there is only 1 cubed root.

(fourth root of 16) = 2 or -2

We know that , so reversing gives: = 2

Also notice: so also = -2

Remember:

Roots are opposite to powers, therefore a power and it’s root undo each other.

e.g. , therefore (we are back to where we started)

The general formula is:

Roots and powers of numbers can be worked out using a calculator, but we need some rules to help us when we have algebra involved.

The Rules of Indices:

The following are the rules that you need to learn and practice:

Rule / What does it mean? / Example
/ If you multiply 2 numbers with the same base you add the powers. /
/ If you divide 2 numbers with the same base you subtract the powers. /
/ If you have a power inside and a power outside of a bracket you multiply the powers. /
/ A negative power means “one over” so everything is sent to the bottom of a fraction. /
/ A fractional power means a root. The bottom of the fraction tells you which root to take and the top tells you which power. /
/ Anything to the power of zero = 1 /
/ Any number to the power of 1 stays the same. /
/ 1 to the power of anything = 1 /

This may seem like a lot to learn but as you practice them they will become easier to remember.

(See the following pages for some examples.)

Examples:

Simplify

The bases are all the same and so we just add the powers:

Simplify

Be careful not to combine different bases. We can only add the powers with a base of . The base of is different so it stays separate.

Simplify

We have numbers as well so just multiply the numbers and add the powers.

Simplify

Divide the numbers and subtract the powers.

Simplify

Subtract the powers but be careful with the signs.

Simplify

Multiply the powers (and remember that 2 is also to the power of 2).

Simplify

Remember the rule that the (–) sign in front of a power sends everything to the bottom.

Simplify

This is the same as: .

Simplify

Now the (–) sign in front of the poweris already on the bottom of the fraction so this time it sends everything to the top.

Write as a power

Remember that roots are fractional powers. This is the square root and so the denominator of the fraction will be 2.

Write as a power

This is the fourth root and so the denominator of the fraction will be 4.

Write as a power

This is the third root and so the denominator of the fraction will be 4. It is then squared and so the numerator will be 2

Simplify