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Lesson · Grade 10 · Foundations of Mathematics and Pre Calculus

Negative exponents: dividing instead of multiplying

  • Grade 10
  • Foundations of Mathematics and Pre Calculus
  • 1 min 54 s
  • British Columbia
  • BC10-MA-01

Starting from 2³ = 2 × 2 × 2 = 8, a table of powers of 2 lowers the exponent one step at a time and halves the value each time, through 2⁰ = 1, to show that 2⁻³ = 1 ÷ 2 ÷ 2 ÷ 2 = 1/2³ = 1/8. It names the result as a reciprocal, x⁻ⁿ = 1/xⁿ for x not 0, checks it with 2³ × 2⁻³ = 8 × 1/8 = 1, flips a fraction base, (2/3)⁻² = (3/2)² = 9/4, crosses out the common mistake 2⁻³ = −8, and ends on the boxed rule.

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In one sentence

The question this lesson answers

What does 2 to the power of negative 3 mean?

Here 2 is the base, and the small raised number, negative 3, is the exponent.

Transcript

What the lesson says

What does 2 to the power of negative 3 mean?

Here 2 is the base, and the small raised number, negative 3, is the exponent.

A positive exponent counts how many copies of the base are multiplied: 2 to the power of 3 is 2 times 2 times 2, which is 8.

Now lower the exponent by 1 each time. There is one fewer 2 to multiply, so the value halves: 4, then 2, then 1.

So 2 to the power of 0 is 1. Below 0 the exponents are negative, and halving gives one half, one quarter and one eighth.

Starting at 1, we divided by 2 three times. That is 1 over 2 to the power of 3, so 2 to the power of negative 3 is one eighth.

1 over a number is called its reciprocal. So x to the power of negative n is 1 over x to the power of n, as long as x is not 0.

Check: 2 to the power of 3 times 2 to the power of negative 3 is 8 times one eighth, which is 1.

To find the reciprocal of a fraction, flip it over. So 2 thirds to the power of negative 2 is 3 halves to the power of 2.

That is 3 times 3 over 2 times 2, which is 9 over 4.

A common mistake is to say 2 to the power of negative 3 is negative 8. The minus sign in the exponent means take the reciprocal, so it is one eighth.

To recap: a negative exponent means divide instead of multiply, so x to the power of negative n is 1 over x to the power of n.

Curriculum

Where this fits

British Columbia, Grade 10

Foundations of Mathematics and Pre Calculus 10, Number: operations on powers with integral exponentsBC10-MA-01

Next: Simplifying with exponent laws and negative exponents

The outcomes and strands follow the British Columbia curriculum.

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