Indices

Lesson

When multiplying a number by itself repeatedly, we are able to use index notation to write the expression more simply. Here we are going to look at a rule that allows us simplify products that involve the multiplication of index terms.

Consider the expression $a^5\times a^3$`a`5×`a`3. Notice that the terms share like bases.

Let's think about what this would look like if we expanded the expression:

We can see that there are eight $a$`a`s being multiplied together, and notice that $8$8 is the sum of the powers in the original expression.

So, in our example above,

$a^5\times a^3$a5×a3 |
$=$= | $a^{5+3}$a5+3 |

$=$= | $a^8$a8 |

Le's look at a specific example. Say we wanted to find the value of $4^2\times4^3$42×43. By evaluating each product separately we would have

$4^2\times4^3$42×43 | $=$= | $16\times64$16×64 |

$=$= | $1024$1024 |

Alternatively, by first expanding the terms in the original expression we can arrive at a simplified version of the expression on our way to the final value.

$4^2\times4^3$42×43 | $=$= | $\left(4\times4\right)\times\left(4\times4\times4\right)$(4×4)×(4×4×4) |

$=$= | $4^5$45 | |

$=$= | $1024$1024 |

Notice in the second line we have identified that $4^2\times4^3=4^5$42×43=45.

We can avoid having to write each expression in expanded form by using the multiplication law.

The multiplication law

For any base number $a$`a`, and any numbers $m$`m` and $n$`n` as powers,

$a^m\times a^n=a^{m+n}$`a``m`×`a``n`=`a``m`+`n`

That is, when multiplying terms with a common base:

- Keep the same base
- Find the sum of the powers

When multiplying terms with like bases, we **add** the indices (or powers).

The multiplication law **only** works for terms with the **same bases**.

Consider the expression $7^2\times3^2$72×32.

$7$7 and $3$3 are not the same base terms, so we cannot simplify this expression any further.

But we can simplify the following expression: $7^2\times7^4\times3^2$72×74×32.

Notice that two of the terms have like bases, so we can add their powers.

$7^2\times7^4\times3^2$72×74×32 | $=$= | $7^{2+4}\times3^2$72+4×32 |

$=$= | $7^6\times3^2$76×32 |

An index (or power) is telling us to multiply the base number by itself a certain number of times. For example:

$5^4$54 | $=$= | $5\times5\times5\times5$5×5×5×5 |

$5^3$53 | $=$= | $5\times5\times5$5×5×5 |

$5^2$52 | $=$= | $5\times5$5×5 |

From this pattern we can see that $5^1$51 is the same as $5$5.

We can use this fact to simplify an expression like $9^2\times9^3\times9$92×93×9 by first writing it as $9^2\times9^3\times9^1$92×93×91 before applying the multiplication law.

$9^2\times9^3\times9$92×93×9 | $=$= | $9^2\times9^3\times9^1$92×93×91 |

$=$= | $9^{2+3+1}$92+3+1 | |

$=$= | $9^6$96 |

Simplify the following, giving your answer with a positive index: $2^2\times2^2$22×22

Simplify the following, giving your answer in index form: $4\times5^6\times5^7$4×56×57.

Simplify the following, giving your answer in index form: $9\times\left(-10\right)^4\times10^8$9×(−10)4×108.