6. Understanding Integer Exponents

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Learning Objective:

Use a positive integer exponent to denote repeated multiplication of a number by itself and a negative integer exponent to denote the reciprocal of a number with a positive integer exponent.

Video:
Interactive Activity

PDF Notes Link:

Understanding Integer ExponentsDownload

Notes:

Positive exponents

The expression “a with exponent n” means “a times a times a, and so on, n times”: an = a x a x … x a. Here a is the base (any real number) and n is the exponent. We can also say “a raised to the power n” or simply “a to the n.” For example, 25 = 2 x 2 x 2 x 2 x 2 = 32.

More examples

\left(\frac{1}{2}\right)^5 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{32}

\left(\frac{1}{2}\right)^5 = \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} = \frac{1}{32}

\pi^5 = \pi \times \pi \times \pi \times \pi \times \pi

Special cases

  • If n = 1, we have a special case with just one factor: a1 = a. For example, 21 = 2.
  • If n = 0: a0 = 1 for all real numbers a except 0 (although in many branches of math, 00 is also defined to be 1). For example, 20 = 1.

Negative exponents

  • If n = -1, a-1 is defined as the reciprocal of a: a-1 = 1/a, for a ≠ 0. For example, 2-1 = 1/2.
  • More generally, a-n is defined as the reciprocal of an: a-n = 1/an, for a ≠ 0. For example, 2-5 = 1/25 = 1/32.
  • Equivalently, a-n = 1/an = (1/a)n. For example, 2-5 = (1/2)5.
  • Similarly, (1/a)-n = an. For example, (1/2)-5 = 25.

More examples

(-2)^{-5} = \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) \times \left(-\frac{1}{2}\right) = -\frac{1}{32}

\pi^{-5} = \frac{1}{\pi} \times \frac{1}{\pi} \times \frac{1}{\pi} \times \frac{1}{\pi} \times \frac{1}{\pi} = \frac{1}{\pi^5}

More complex expressions

With more complex expressions it can be simpler to use a “dot” to represent multiplication:

  • ban = b . an ≠ (ba)n. For example, 3 . 25 = 3 . 32 = 96. However, (3 . 2)5 = 65 = 7776.
  • ba-n = b . a-n = b . 1/an = b/an. For example, 3 . 2-5 = 3 . 1/25 = 3/25 = 3/32.
  • b/a-n = b . 1/a-n = b . (1/a)-n = ban. For example, 3/2-5 = 3 . 1/2-5 = 3 . (1/2)-5 = 3 . 25 = 3 . 32.

More examples

4(-2)^{-3} = \frac{4}{(-2)^3} = \frac{4}{(-2)\times(-2)\times(-2)} = \frac{4}{-8} = -\frac{1}{2}

\frac{-3}{(-4)^{-2}} = -3(-4)^2 = -3\cdot 16 = -48

Transcript:

The video transcripts are accessible for viewing and downloading below.

Understanding-Integer-ExponentsDownload

Video Link:

https://media.tru.ca/media/Math+06/0_rl8m4pdd

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