Indices

Lesson

We've already learnt about how to use scientific notation to write really big or really small numbers. Remember they are written in the form $a\times10^b$`a`×10`b`. Since these numbers are all written in relation to a power of 10, numbers written in scientific notation always have indices (or powers). As such, we can simplify expressions written in scientific notation using index laws, such as the multiplication law or the division law.

If you are adding or subtracting numbers written in scientific notation, you need to make sure that the powers of ten are the same. To do this, you may need to factorise some of the powers of 10 based on the rules of the multiplication law. Then you can use the decimal number, rather than the number scientific notation and solve the problem. The process goes:

$a\times10^n\pm b\times10^m$a×10n±b×10m |
$=$= | $a\times10^n\pm b\times10^{m-n}\times10^n$a×10n±b×10m−n×10n |

$=$= | $\left(a\pm b\times10^{m-n}\right)\times10^n$(a±b×10m−n)×10n |

Tip

Make sure you check that your final answer is expressed appropriately in scientific notation.

Use index laws to simplify $2\times10^6\times6\times10^5$2×106×6×105. Give your answer in scientific notation.

**Think:** Let's simplify the expression first. Remember to express our answer in scientific notation, we'll need to express the coefficient as a value between $1$1 and $10$10 and then multiply it by the correct power of $10$10.

**Do:**

$2\times10^6\times6\times10^5$2×106×6×105 | $=$= | $12\times10^{6+5}$12×106+5 |

$=$= | $12\times10^{11}$12×1011 |

$12$12 can be as expressed as $1.2\times10^1$1.2×101. We will use this to write our answer in scientific notation.

$1.2\times10^1\times10^{11}$1.2×101×1011 | $=$= | $1.2\times10^{1+11}$1.2×101+11 |

$=$= | $1.2\times10^{12}$1.2×1012 |

Use index laws to simplify $\frac{4\times10^{-5}}{16\times10^4}$4×10−516×104. Give your answer in scientific notation.

Evaluate $4.28\times10^6+3.34\times10^7$4.28×106+3.34×107.

Give your answer in scientific notation to three significant figures.

Extend powers to include integers and fractions.

Apply numeric reasoning in solving problems