Powers, roots and index laws

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Notes de leçon

Squares, Cubes and Higher Powers

  • A power is a number multiplied by itself a certain number of times.
  • The base is the number being multiplied; the exponent (or index) tells you how many times to multiply it.
  • For example, bn means b imes b imes \dots imes b with n copies of b.
  • b2 is read as 'b squared' because it gives the area of a square with side length b.
  • b3 is read as 'b cubed' because it gives the volume of a cube with side length b.
  • Higher powers like b4, b5 follow the same pattern: multiply the base by itself that many times.

Index Notation

  • We write powers using a superscript: bn, where b is the base and n is the index.
  • The index is also called the exponent or power.
  • You may see the caret symbol used instead: bn.
  • The expression bn is read as 'b to the power of n' or 'b to the n'.
  • The result of a power calculation is also called a power.

Square Roots and Cube Roots

  • The square root of a number is the value that, when squared, gives that number.
  • The square root of x is written \√{x}.
  • For example, \√{25} = 5 because 52 = 25.
  • The cube root of a number is the value that, when cubed, gives that number.
  • The cube root of x is written \√[3]{x}.
  • For example, \√[3]{27} = 3 because 33 = 27.
  • Roots can be estimated: \√{50} is between 7 and 8 because 72 = 49 and 82 = 64.

Multiplying Powers with the Same Base

  • When multiplying powers with the same base, add the indices: am imes an = am+n.
  • For example, 23 imes 24 = 23+4 = 27.
  • This works because you are combining the repeated multiplications.
  • The base stays the same; only the index changes.

Dividing Powers with the Same Base

  • When dividing powers with the same base, subtract the indices: am \div an = am-n.
  • For example, 56 \div 52 = 56-2 = 54.
  • Again, the base remains the same.
  • This rule works when the first index is larger than the second.

Powers of a Power

  • To raise a power to another power, multiply the indices: (am)n = am imes n.
  • For example, (32)4 = 32 imes 4 = 38.
  • This is because you are multiplying the base by itself m imes n times in total.

The Zero Index

  • Any non-zero number raised to the power 0 equals 1: a0 = 1.
  • For example, 70 = 1 and (-3)0 = 1.
  • This follows from the division rule: an \div an = an-n = a0, and any number divided by itself is 1.

Negative Indices

  • A negative index means the reciprocal of the positive power: a-n = rac{1}{an}.
  • For example, 2-3 = rac{1}{23} = rac{1}{8}.
  • Also, a-1 = rac{1}{a}.
  • Negative indices do not make the result negative; they make it a fraction.

Powers of 10

  • Powers of 10 are easy to calculate: 10n is a 1 followed by n zeros.
  • For example, 103 = 1000.
  • Negative powers of 10 give decimals: 10-1 = 0.1, 10-2 = 0.01, 10-3 = 0.001.
  • Powers of 10 are useful for writing very large or very small numbers in standard form.

Diapos

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  1. 1.In the expression bn, what is the number n called?

    Easy
    • Athe base
    • Bthe exponent
    • Cthe coefficient
    • Dthe product

  2. 2.In the expression bn, the number b is called the base.

    Easy

    True or false?

  3. 3.Which of the following are correct names for the exponent in a power? (select all that apply)

    Easy
    • Aindex
    • Bpower
    • Cbase
    • Dcoefficient
    • Eroot

  4. 4.Which expression is equivalent to 5 × 5 × 5 × 5?

    Easy
    • A53
    • B54
    • C45
    • D5 × 4

  5. 5.What is the value of 34?

    Medium
    • A12
    • B64
    • C81
    • D27

  6. 6.What is the value of 60?

    Medium
    • A0
    • B1
    • C6
    • D60

  7. 7.Simplify 75 × 73, leaving your answer as a power of 7.

    Medium
    • A78
    • B715
    • C72
    • D498

  8. 8.Simplify 89 ÷ 84, leaving your answer as a power of 8.

    Medium
    • A813
    • B85
    • C836
    • D15

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