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By the end of this topic, you should be able to:
When you multiply the same number by itself several times, you can write it in a shorter way using index notation.
For example:
In this example:
The index is also called a power or an exponent. All three words mean the same thing.
We read 2⁴ as "2 to the power of 4" or simply "2 to the 4".
When the index is a positive whole number, it tells you how many times to multiply the base by itself.
Examples:
A number is squared when it is multiplied by itself once. We use the symbol ² for squared.
Examples:
The square root is the opposite operation. It asks: "What number was multiplied by itself to get this answer?"
We use the symbol √ for square root.
Examples:
Important note: The square root symbol √ only gives the positive answer. So √25 = 5, not -5, even though (-5) × (-5) also equals 25.
A number is cubed when it is multiplied by itself twice. We use the symbol ³ for cubed.
Examples:
The cube root is the opposite operation. It asks: "What number was multiplied by itself twice to get this answer?"
We use the symbol ∛ for cube root.
Examples:
The laws of indices are rules that help you work with powers more easily. Instead of writing out all the multiplications, you can use these shortcuts.
When you multiply powers with the same base, add the indices.
Formula: a^m × a^n = a^(m+n)
Examples:
Why does this work? Look at 2³ × 2²:
You have 3 twos plus 2 more twos, making 5 twos in total.
When you divide powers with the same base, subtract the indices.
Formula: a^m ÷ a^n = a^(m-n)
Examples:
Why does this work? Look at 2⁵ ÷ 2²:
Two of the twos cancel out, leaving 3 twos.
When you raise a power to another power, multiply the indices.
Formula: (a^m)^n = a^(m×n)
Examples:
Why does this work? Look at (2³)²:
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