Algebra

Indices & the Law of Indices

Introduction

Indices are a useful way of more simply expressing large numbers. They also present us with many useful properties for manipulating them using what are called the Law of Indices.

What are Indices?

The expression \(2^{5}\) is defined as follows:

We call "2" the base and "5" the index.

Law of Indices

To manipulate expressions, we can consider using the Law of Indices. These laws only apply to expressions with the same base, for example, \(3^{4}\) and \(3^{2}\) can be manipulated using the Law of Indices, but we cannot use the Law of Indices to manipulate the expressions \(3^{5}\) and \(5^{7}\) as their base differs (their bases are 3 and 5, respectively).

Six rules of the Law of Indices

Any number, except 0, whose index is 0 is always equal to 1, regardless of the value of the base.

Example

Simplify \(2^{0}\)

Solution

\[ 2^{0} = 1 \]

Example

Simplify \(2^{-2}\)

Solution

1. Step 1 — Apply the negative-index law \(a^{-m} = \frac{1}{a^{m}}\)
   \[ 2^{-2} = \frac{1}{2^{2}} \]
2. Step 2 — Evaluate

   Since \(2^{2} = 4\):

   \[ 2^{-2} = \frac{1}{4} \]

   That is, 2 to the power minus 2 is one quarter.

To multiply expressions with the same base, copy the base and add the indices.

Example

Simplify \(5 \times 5^{3}\)

(note: \(5 = 5^{1}\))

Solution

1. Step 1 — Apply the multiplication law \(a^{m} \times a^{n} = a^{m+n}\)

   Write \(5\) as \(5^{1}\), so both factors are powers of \(5\), then add the indices \(1\) and \(3\):

   \[ 5^{1} \times 5^{3} = 5^{1+3} \]
2. Step 2 — Add the indices
   \[ = 5^{4} \]
3. Step 3 — Evaluate

   \(5^{4}\) is four factors of \(5\) multiplied together:

   \[ 5 \times 5^{3} = 5 \times 5 \times 5 \times 5 = 625 \]

   That is, 5 times 5 cubed is 625.

To divide expressions with the same base, copy the base and subtract the indices.

Example

Simplify \(5\left(y^{9} \div y^{5}\right)\)

Solution

1. Step 1 — Apply the division law \(a^{m} \div a^{n} = a^{m-n}\)

   Inside the bracket, subtract the index \(5\) from the index \(9\):

   \[ 5\left(y^{9} \div y^{5}\right) = 5\left(y^{9-5}\right) \]
2. Step 2 — Subtract the indices
   \[ 5\left(y^{9} \div y^{5}\right) = 5y^{4} \]

   That is, the expression is 5 times \(y\) to the power 4.

To raise an expression to the nth index, copy the base and multiply the indices.

Example

Simplify \(\left(y^{2}\right)^{6}\)

Solution

1. Step 1 — Apply the power-of-a-power law \(\left(a^{m}\right)^{n} = a^{mn}\)

   Multiply the index \(2\) by the index \(6\):

   \[ \left(y^{2}\right)^{6} = y^{2 \times 6} \]
2. Step 2 — Multiply the indices
   \[ \left(y^{2}\right)^{6} = y^{12} \]

   That is, the expression is \(y\) to the power 12.

Two further laws follow from the laws above and are used throughout this topic's questions. To raise a product to an index, raise each factor to that index:

Similarly, to raise a quotient to an index, raise the numerator and the denominator to that index:

Example

Simplify \(125^{2/3}\)

Solution

1. Step 1 — Apply the fractional-index law \(a^{m/n} = \sqrt[n]{a^{m}} = \left(\sqrt[n]{a}\right)^{m}\)

   The denominator \(3\) is the root and the numerator \(2\) is the power, so take the cube root first:

   \[ 125^{2/3} = \left(\sqrt[3]{125}\right)^{2} \]
2. Step 2 — Take the cube root

   Recognise that the cube root of \(125\) is \(5\), since \(5^{3} = 125\):

   \[ = 5^{2} \]
3. Step 3 — Square the result
   \[ 125^{2/3} = 25 \]

   That is, 125 to the power 2 over 3 is 25.

You have now learnt the important rules of the Law of Indices and are ready to try out some examples!