3
Simplifying expressions with variable exponents and solving problems
Mathematics Prep 2 Term 1, Unit 1, Lesson 5
GoalSimplify expressions with n in the exponent, find their value for a given n, and solve problems with powers.
New words
variable exponent · an exponent that contains a letter, such as 3ⁿ⁻¹simplify · write with fewer powers, using the lawscommon base · write every power with the same base, such as 2volume of a cube · edge cubed: V = a³
Explanation
The laws work when the exponent is a letter. To simplify, give every power the same base, then add exponents when multiplying and subtract when dividing. Example: 6ⁿ × 3ⁿ⁻¹ × 2⁻ⁿ. Write 6ⁿ = 2ⁿ × 3ⁿ. Then 2ⁿ × 2⁻ⁿ = 2⁰ = 1, and what is left is 3ⁿ × 3ⁿ⁻¹ = 3²ⁿ⁻¹. For n = 2 this is 3³ = 27, and checking the original: 6² × 3¹ × 2⁻² = 36 × 3 ÷ 4 = 27. Sometimes all the n disappear: (4ⁿ × 2ⁿ⁺¹) ÷ 8ⁿ = 2²ⁿ⁺ⁿ⁺¹⁻³ⁿ = 2¹ = 2 for every n. Powers also give volumes: a cube with edge a has volume a³.
| To simplify powers… | Do this |
|---|---|
| same base, multiply | add the exponents |
| same base, divide | subtract the exponents |
| power of a power | multiply the exponents |
| different bases | write them with a common base |
Worked examples
Simplify 2ⁿ⁺² − 2ⁿ and find its value for n = 3. 2ⁿ⁺² = 2ⁿ × 4, so 2ⁿ × 4 − 2ⁿ = 2ⁿ(4 − 1) = 3 × 2ⁿ. For n = 3: 3 × 8 = 24. Check: 2⁵ − 2³ = 32 − 8 = 24.
Simplify 6ⁿ × 3ⁿ⁻¹ × 2⁻ⁿ. As above it is 3²ⁿ⁻¹. For n = 3 it is 3⁵ = 243. Check: 6³ × 3² × 2⁻³ = 216 × 9 ÷ 8 = 243.
A cube has edge 2√3 cm. Its volume is (2√3)³ = 2³ × (√3)³ = 8 × 3√3 = 24√3 cm³, since (√3)³ = (√3)² × √3 = 3√3.
Class activity
Choose your own n. Pick n = 1, 2, 3 and 4. Find the values of 2ⁿ⁺² − 2ⁿ and 3 × 2ⁿ for each n. Do they always agree? Then try (4ⁿ × 2ⁿ⁺¹) ÷ 8ⁿ for n = 1 and n = 5.
Practice
1
Simplify 2ⁿ × 2ⁿ⁺¹ and find its value for n = 2.
2²ⁿ⁺¹. For n = 2: 2⁵ = 32.
2
Simplify 3ⁿ⁺² ÷ 3ⁿ.
3² = 9.
3
Simplify (5ⁿ)² × 5⁻ⁿ and find its value for n = 3.
5²ⁿ × 5⁻ⁿ = 5ⁿ. For n = 3: 125.
4
A cube has edge 3√2 cm. Find its volume.
(3√2)³ = 27 × (√2)³ = 27 × 2√2 = 54√2 cm³.
5
Write 4 × 10⁻² as a decimal.
4 × 0.01 = 0.04.