Lesson 1: Exponents and Exponent Laws
Lesson 1: Exponents and Exponent Laws
What exponents mean, the exponent laws, zero and negative exponents.
Exponents are very important and powerful. Read this blog post based on an ancient Indian legend about the creator of chess.
This lesson teaches about exponents. An exponent shows how many times the number needs to be multiplied by itself. For example, 5³ = 5 × 5 × 5 = 125.
In the order of operations, BEDMAS, exponents come right after operations in brackets.
Sometimes exponents are used with variables. Then, exponent laws are used in order to simplify expressions with variables.
This lesson explains how to use general exponent laws and provides insight into operations with exponents.
A negative exponent means the reciprocal of the power: a−n = 1/an. For example, 2−3 = 1/23 = 1/8.
Any number (except 0) or an algebraic term raised to the exponent of 0 is equal to 1.
When raising a fraction to an exponent it is necessary to raise both the numerator and the denominator to that exponent separately.
Exponents and Exponent Laws Short Animation
Key ideas: Exponents and Exponent Laws
A power has a base and an exponent. In 53, the base is 5 and the exponent 3 tells us how many times to multiply the base by itself: 53 = 5 × 5 × 5 = 125. In BEDMAS, exponents are evaluated right after brackets.
The exponent laws are shortcuts that work when the bases are the same (a ≠ 0, b ≠ 0):
- Product law: am × an = am + n (multiply → add the exponents)
- Quotient law: am ÷ an = am − n (divide → subtract the exponents)
- Power of a power: (am)n = amn (multiply the exponents)
- Power of a product: (ab)n = anbn
- Power of a quotient: (a/b)n = an/bn (raise the numerator and the denominator)
- Zero exponent: a0 = 1
- Negative exponent: a−n = 1/an (take the reciprocal and make the exponent positive)
Worked examples
Example 1
Simplify (2x3)(5x4).
- Multiply the coefficients: 2 × 5 = 10.
- Same base x, so add the exponents: x3 + 4 = x7.
- Answer: 10x7
Example 2
Simplify (3a2b)3 ÷ (9a4b2).
- Power of a product: (3a2b)3 = 27a6b3.
- Divide coefficients: 27 ÷ 9 = 3. Subtract exponents: a6 − 4 = a2 and b3 − 2 = b.
- Answer: 3a2b
Example 3
Evaluate 2−3 + 40.
- 2−3 = 1/23 = 1/8, and 40 = 1.
- 1/8 + 1 = 9/8.
- Answer: 9/8 (or 1⅛)
Example 4
Simplify x5y−2 ÷ (x2y3) and write the answer with positive exponents.
- x5 − 2 = x3 and y−2 − 3 = y−5.
- y−5 = 1/y5.
- Answer: x3/y5
Common mistakes
- Multiplying the exponents when multiplying powers: x3 × x4 is x7, not x12.
- Thinking a negative exponent makes the answer negative: 2−3 = 1/8, not −8.
- Mixing up −32 and (−3)2: −32 = −9 (only the 3 is squared), but (−3)2 = 9.
- Using the laws with different bases: 23 × 32 is 8 × 9 = 72, not 65.
- Distributing an exponent over addition: (a + b)2 is not a2 + b2.
Questions students ask
Why does any number to the power of zero equal 1?
Because an ÷ an = 1 (anything divided by itself), and by the quotient law it also equals an − n = a0. So a0 = 1, as long as a is not 0.
What is the difference between −24 and (−2)4?
Exponents come before the negative sign in BEDMAS unless there are brackets. −24 = −16, while (−2)4 = 16.
Grade 9 Exponents Quiz
Check your understanding
Try the quiz for this lesson. You’ll see right away if each answer is correct.