Exponent Rules

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

In short: Exponent rules are shortcuts for merging powers of one base: stacking factors adds their exponents, splitting them subtracts, and a tower of two exponents collapses into a product.

8th GradeNumbers and Operations8.EE.A.1

What is Exponent Rules?

The exponent rules are shortcuts for combining powers without writing out every factor. They exist because \(a^n\) means \(n\) copies of \(a\) multiplied together, so merging copies simply adds or removes them.

Three laws cover most work with a nonzero base \(a\): the product rule \(a^m \cdot a^n = a^{m+n}\), the quotient rule \(\frac{a^m}{a^n} = a^{m-n}\) and the power rule \((a^m)^n = a^{mn}\). Each one needs the same base in the terms being combined.

How to spot it

  • An expression multiplies or divides powers sharing one base, like five to the sixth times five squared.
  • A power sits in parentheses and wears a second exponent outside, as in the cube of a square.
  • A long string of repeated factors begs for a compact form with a single exponent.
  • Bases differ, so you must decide whether any rule applies before touching the exponents.

The method

  1. Look at the bases first; when they differ, try rewriting one, or leave the factors as they are.
  2. Two powers side by side with a times sign: the base stays, the exponents are added.
  3. A power over a power: the base stays, and the exponent below is taken away from the exponent above.
  4. A power inside parentheses carrying its own exponent: the base stays, and the two exponents are multiplied. Evaluate at the end if a number is wanted.
PRODUCTa^m × a^n= a^(m+n)5^6 × 5^2 = 5^8QUOTIENTa^m ÷ a^n= a^(m−n)5^6 ÷ 5^4 = 5^2POWER(a^m)^n= a^(m×n)(5^2)^3 = 5^6
Add exponents to multiply, subtract to divide, multiply to raise a power to a power.

Exponent Rules: solved example

Problem

Simplify \(\frac{5^6 \cdot 5^2}{5^4}\) and give its value.

Solution

Top first, by the product rule: \(5^6 \cdot 5^2 = 5^{8}\).

Then divide: \(\frac{5^8}{5^4} = 5^{8-4} = 5^4\). Since \(5^4 = 625\), the value is \(625\).

Try it

Question 1

Simplify \((3^2)^4\).

Show the answer

Multiply the exponents: \(3^{2 \cdot 4} = 3^8 = 6{,}561\).

Question 2

Simplify \((4x^3)^2\).

Show the answer

Square each factor: \(4^2 \cdot (x^3)^2 = 16x^6\).

Exponent Rules vs. Scientific Notation

Exponent rules are the algebra laws for combining powers of any base. Scientific notation is a way of writing numbers using one specific base, 10. You use the rules whenever you multiply or divide numbers in scientific notation, adding or subtracting the powers of ten.

Scientific Notation

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