If you work through this section you should be able to:
A power or an exponent of a number states how many times to multiply the number by itself. It is written as a small superscripted number on the top right of a number, for example 25 = 2×2×2×2×2. The superscript 5 is called a power or index and the number 2 is called the base. It is pronounced “two raised to the power of five”.
Raising to the power of 2 is called squaring: “4 squared” is 42 = 4 × 4 = 16;
and to the power of 3 is called cubing: “5 cubed” is 53 = 5 × 5 × 5 = 125.
Note: Any number raised to the power of 1 is itself, e.g. 61 = 6.
Any number raised to the power of 0 has the value to 1, e.g. 90 = 1.
According to the BEDMAS rule, exponentiation is done after brackets and before multiplication.
So: 3 × 23 = 3 × 8 = 24 — but (3 × 2)3 = 63 = 216.
Negative powers denote the reciprocal. The positive powers are calculated first then the reciprocal is taken.
So:
If then is the root of . For example, if 62 = 36, then 6 is the second root of 36. The second root is usually called the square root. It is written as 6 = .
The third root is usually called the cube root. For example, 3 = .
Note also that when −6 is squared we again obtain 36, that is (−6)2 = 36. This means that 36 has another square root, −6. Therefore, = ±6
Fractional powers can be written as . A fractional power can be broken into two parts: power and root. You can either do the power first then take the root, or alternatively take the root first and then do the power. Therefore:
= =
So: = = 33 = 27
The rules of indices can be used to manipulate powers related expressions. The rules of indices are given as below:
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You can download a version of this Powers activity in Word format:
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