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Skills for Learning

Maths & Stats - Powers and Logarithms

Learning Outcomes

If you work through this section you should be able to:

  • Understand what is meant by a power or index.
  • Evaluate expressions involving powers.
  • Evaluate expressions using the rules of indices.
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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.

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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.

Examples

  1.   −42 = −4 × 4 = −16
  2.   (−4)2 = (−4) × (−4) = 16
  3.   2(3 × 52 − 8) = 2(3 × 25 − 8)
      = 2 (75 − 8)
      = 134
  4.   3(6 + 2 × 32)2 = 3(6 + 2 × 9)2
      = 3(6 + 18)2
      = 3(24)2
      = 3 × 576
      = 1728

Negative powers denote the reciprocal. The positive powers are calculated first then the reciprocal is taken.

So: 3-2 = 132=19

If xa = b then x is the ath root of b. 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 = 36.

The third root is usually called the cube root. For example, 3 = 273.

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, 36 = ±6

Fractional powers can be written as amn. A fractional power can be broken into two parts: mth power and nth root. You can either do the power first then take the root, or alternatively take the root first and then do the power. Therefore:

amn = amn = an m

So: 8134 = (814)3 = 33 = 27

Examples

  •  643 = 433 = 4
  •  25-12 = 125 = 15 = 0.2
  •  823 = 832 = 22 = 4
  •  125-23 = 112532 = 152 = 125 = 0.04

The rules of indices can be used to manipulate powers related expressions. The rules of indices are given as below:

an  x  am=an+m

an÷am=an-m

anm=anm

a-n=1an

a1n=an

anm=a1mn=amn

Examples

  1.   43 x 45 = 43+5 = 48
  2.   X2 x X7 ÷ X4 = X2+7-4
      = X5
  3.   (35)2 = 310
  4.   s3 ÷ t−4 × (s−3t−2)3 = s3 ÷ t−4 × s−9 × t−6
      = s3−9 × t4−6
      = s−6t−2
  5.   823÷8 = 8231
       813
      = 183
      = 12
      = 0.5

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