The Laws of Indices

Introduction to Indices

An index or power is a small number placed above and to the right of a base number. The index indicates how many times the base number should be multiplied by itself.

For example, 2\times 2\times 2\times 2\times 2 can be written as { 2 }^{ 5 }. The base is 2, and the exponent is 5. It is read as “2 to the power of 5.”

Let’s look at some more examples:

  • 3\times 3={ 3 }^{ 2 }
  • 5\times 5\times 5\times 5={ 5 }^{ 4 }
  • 7\times 7\times 7={ 7 }^{ 3 }

Laws of Indices

Multiplying indices

When multiplying indices with the same base, we add the powers:

a^{b}\times a^{c}=a^{b+c}

Examples:

  • 3^{ 7 }\times { 3 }^{ 4 }=3^{ 7+4 }={ 3 }^{ 11 }
  • { 5 }^{ 8 }\times { 5 }^{ 2 }={ 5 }^{ 10 }
  • { 3 }^{ 3 }\times { 3 }^{ 6 }={ 3 }^{ 9 }

Dividing indices

When dividing indices with the same base, we subtract the powers:

\dfrac{a^{b}}{a^{c}}=a^{b-c}

Examples:

  • { 8 }^{ 14 }\div{ 8 }^{ 7 }={ 8 }^{ 14-7 }={ 8 }^{ 7 }
  • { 2 }^{ 10 }\div{ 2 }^{ 4 }={ 2 }^{ 10-4 }={ 2 }^{ 6 }
  • { 3 }^{ 7 }\div { 3 }^{ 5 }={ 3 }^{ 7-5 }={ 3 }^{ 2 }
  • { 5 }^{ 20 }\div { 5 }^{ 12 }={ 5 }^{ 20-12 }={ 5 }^{ 8 }

Power of a power

When raising an index to another power, we multiply the powers:

\left( a^{b}\right) ^{c}=a^{b\times c}

Examples:

  • ({ { 3 }^{ 2 }) }^{ 4 }={ 3 }^{ 2 }\times { 3 }^{ 2 }\times { 3 }^{ 2 }\times { 3 }^{ 2 }={ 3 }^{ 2+2+2+2 }=3^{8}
  • ({ { 2 }^{ 5 }) }^{ 2 }={ 2 }^{ 5\times 2 }={ 2 }^{ 10 }
  • ({ { 3 }^{ 7 }) }^{ 2 }={ 3 }^{ 7\times 2 }={ 3 }^{ 14 }
  • ({ { 5 }^{ 10 }) }^{ 10 }={ 5 }^{ 10\times 10 }={ 5 }^{ 100 }
  • ({ { 4 }^{ 3 }) }^{ 5 }={ 4 }^{ 3\times 5 }={ 4 }^{ 15 }

Power of a product

When raising a product to a power, we distribute the power to each factor:

\left( ab\right) ^{n}=a^{n}\times b^{n}

Examples:

\left( 4\times 3\right) ^{2}=4^{2}\times 3^{2}

\left( 5\times 7\right) ^{3}=5^{3}\times 7^{3}

Power of zero

Any non-zero number raised to the power of 0 is equal to 1:

a^{0}=1 (where a\neq 0)

Examples:

  • { 5 }^{ 0 }=1
  • { 8 }^{ 0 }=1
  • { 1000 }^{ 0 }=1,\quad ({ -3) }^{ 0 }=1

Negative indices

A negative index represents the reciprocal of the positive index:

a^{-n}=\dfrac{1}{a^{n}}

Examples:

  • 2^{-3}=\dfrac{1}{2^{3}}=\dfrac{1}{8}
  • 4^{-4}=\dfrac{1}{4^{4}}=\dfrac{1}{256}

Using the Laws of Indices to Simplify Numbers and Equations

We can use the laws of indices to simplify numbers and calculations. For example:

If A=2.3\times 10^{5}, B=4.5\times 10^{-2}, then AB=\left( 2.3\times 10^{5}\right) \times \left( 4.5\times 10^{-2}\right)

=10.35\times 10^{5-2}

=10.35\times 10^{3}

=1.035\times 10\times 10^{3}=1.035\times 10^{4}

Let’s look at two more examples:

Example

Simplify:

1. 3^{5}\times 3^{2}

2. 3^{5}\div 3^{3}

3. \left( 3^{2}\right) ^{5}

4. 3^{-4}

5. 3^{0}

6. \left( 3\times 7\right) ^{2}

1. 3^{5}\times 3^{2}=3^{5+2}=3^{7}3^{5}\times 3^{2}=3^{5+2}=3^{7}

2. 3^{5}\div 3^{3}=3^{5-3}=3^{2}=93^{5}\div 3^{3}=3^{5-3}=3^{2}=9

3. \left( 3^{2}\right) ^{5}=3^{2\times 5}=3^{10}\left( 3^{2}\right) ^{5}=3^{2\times 5}=3^{10}

4. 3^{-4}=\dfrac{1}{3^4}=\dfrac{1}{81}3^{-4}=\dfrac{1}{3^4}=\dfrac{1}{81}

5. 3^{0}=13^{0}=1

6. \left( 3\times 7\right) ^{2}=3^{2}\times 7^{2}=9\times 49=441\left( 3\times 7\right) ^{2}=3^{2}\times 7^{2}=9\times 49=441


Example

Simplify \left( 5^{3}\right) ^{2}\times \left( 5^{-1}\right) ^{4}

\left( 5^{3}\right) ^{2}\times \left( 5^{-1}\right) ^{4}=5^{6}\times 5^{-4}\left( 5^{3}\right) ^{2}\times \left( 5^{-1}\right) ^{4}=5^{6}\times 5^{-4}

=5^{6-4}=5^{6-4}

=5^{2}=5^{2}

=25=25

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