Laws of Exponents
Rules for multiplying, dividing and raising powers.
Key idea
Rules for multiplying, dividing and raising powers.
In depth
With the same base: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ. They follow from counting how many factors of a you have.
Worked example
2³ × 2⁴ = 2⁷ = 128, since (2·2·2)(2·2·2·2) has 7 twos.
Formula
Real-world uses
- Simplifying scientific formulas, compound growth models and computer algorithm running times
Common mistake
Adding exponents when multiplying different bases. 2³ × 3⁴ cannot be combined.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Laws of Exponents?
Rules for multiplying, dividing and raising powers. It is a topic at the advanced level in Arithmetic & Number Sense.
What is the formula for Laws of Exponents?
The key formula is aᵐ·aⁿ = aᵐ⁺ⁿ; (aᵐ)ⁿ = aᵐⁿ.
Where is Laws of Exponents used in real life?
Real-world examples: Simplifying scientific formulas, compound growth models and computer algorithm running times.
What should I learn before Laws of Exponents?
Make sure you are comfortable with Rationalizing Denominators, the previous topic in Arithmetic & Number Sense.
What is a common mistake with Laws of Exponents?
Adding exponents when multiplying different bases. 2³ × 3⁴ cannot be combined.
How can I practise Laws of Exponents?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Laws of Exponents?
Move on to Negative and Zero Exponents next.
Related advanced topics in Arithmetic & Number Sense
- Repeating Decimals – Decimals with a repeating block are rational and convertible to fractions.
- Compound Interest – Interest earned on principal and past interest.
- Inverse Proportion – When one quantity doubles, the other halves.
- Rational Numbers – Numbers expressible as p/q with integers p, q, q≠0.
- Irrational Numbers – Real numbers that cannot be written as a fraction, such as √2 and π.
- Real Numbers – All rational and irrational numbers; the continuous number line.