Negative Exponent

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Negative Exponent: meaning, method and solved example (8th Grade) – Maths PDF

In short: A negative exponent sends the base to the other side of a fraction bar, so a raised to minus n becomes one divided by a raised to plus n, and the value shrinks toward zero.

8th GradeNumbers and Operations8.EE.A.1

What is Negative Exponent?

Despite the minus sign, a negative exponent never makes the result negative; it instead moves the base across the fraction bar. Written out, \(a^{-n} = \frac{1}{a^n}\) for \(a \neq 0\), so \(4^{-3} = \frac{1}{4^3} = \frac{1}{64}\).

Powers of ten show the reason. Dropping the exponent by one divides by 10 each time: \(10^2 = 100\), \(10^1 = 10\), \(10^0 = 1\), \(10^{-1} = 0.1\), \(10^{-2} = 0.01\). Pushing the pattern past zero yields tenths, hundredths and ever smaller pieces.

How to spot it

  • A small raised number wears a minus sign, like seven to the power of negative two.
  • A tiny length, such as a micrometer, appears with a negative power of ten.
  • Dividing powers leaves the bigger exponent on the bottom.
  • The final answer is supposed to show positive exponents only.

The method

  1. Ignore the sign of the base for a moment and focus on the minus in the exponent.
  2. Slide that base across the fraction bar and flip the exponent to positive.
  3. Work out the power now sitting below, then write the fraction or its decimal.
  4. Sanity check: a base above 1 with a minus exponent always lands strictly between 0 and 1.
10^31,00010^210010^11010^0110^-10.1÷ 10÷ 10÷ 10÷ 1010^-2 = 1/10^2 = 1/100 = 0.01
Each step down in the exponent divides by 10, so 10^-1 is 0.1 and 10^-2 is 0.01.

Negative Exponent: solved example

Problem

Compute \(4^{-3}\) and also \(\left(\frac{2}{3}\right)^{-2}\).

Solution

First: \(4^{-3} = \frac{1}{4^3} = \frac{1}{64}\).

Second: a negative exponent flips the fraction, so \(\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^{2} = \frac{9}{4}\). The values are \(\frac{1}{64}\) and \(\frac{9}{4}\).

Try it

Question 1

Express \(5^{-2}\) both ways, fraction and decimal.

Show the answer

\(5^{-2} = \frac{1}{25} = 0.04\).

Question 2

Simplify \(3^2 \cdot 3^{-5}\) using only a positive exponent.

Show the answer

Add exponents: \(3^{-3} = \frac{1}{3^3} = \frac{1}{27}\).

Negative Exponent vs. Zero Exponent

A negative exponent yields a reciprocal, so a base above 1 gives a fraction below 1. A zero exponent yields exactly 1 for any nonzero base. Neither flips the sign of the base itself; that sign belongs to the number, not to the exponent.

Zero Exponent

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