MATHEMATICS HIP-HOP ACADEMY · CHAPTER 01
Numerical Concepts
Part 1 — Indices and Fractional Powers
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About this lesson
This lesson introduces indices and fractional powers. It is Part 1 of the proposed Mathshub Numerical Concepts series for A-Level Senior Five. Read the explanations carefully, study the worked examples, and attempt the exercises before checking the answers.
Learning outcomes
Apply the laws of indices.
Simplify expressions involving zero and negative indices.
Work with fractional indices.
Simplify algebraic powers.
Solve basic exponential equations.
Introduction
This lesson introduces indices and fractional powers. It is Part 1 of the proposed Mathshub Numerical Concepts series for A-Level Senior Five. Read the explanations carefully, study the worked examples, and attempt the exercises before checking the answers.
| Learning outcomes: By the end of this part, you should be able to apply the laws of indices, simplify expressions involving zero and negative indices, work with fractional indices, simplify algebraic powers, and solve basic exponential equations. |
1. Meaning of indices
An index (plural: indices), also called an exponent or power, tells us how many times a base is used as a factor in repeated multiplication.
2⁴ = 2 × 2 × 2 × 2 = 16
In 2⁴, the number 2 is the base and 4 is the index. In general, aⁿ means that a is multiplied by itself n times when n is a positive integer.
2. The laws of indices
For non-zero bases where the expressions are defined, the following laws help simplify powers.
| Rule | Law | Example |
|---|---|---|
| Multiplication, same base | aᵐ × aⁿ = aᵐ⁺ⁿ | x² × x⁵ = x⁷ |
| Division, same base | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | y⁸ ÷ y³ = y⁵ |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (x³)⁴ = x¹² |
| Power of a product | (ab)ⁿ = aⁿbⁿ | (2x)³ = 8x³ |
| Power of a quotient | (a/b)ⁿ = aⁿ/bⁿ | (x/2)² = x²/4 |
| Zero index | a⁰ = 1 | 7⁰ = 1 |
| Negative index | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| Fractional index | a¹/ⁿ = ⁿ√a | 25¹/² = √25 = 5 |
| General fractional index | aᵐ/ⁿ = ⁿ√(aᵐ) | 8²/³ = (∛8)² = 4 |
3. Multiplying and dividing powers
Example 1 — Simplify 3⁴ × 3².
Step 1: The bases are the same, so add the indices.
3⁴ × 3² = 3⁴⁺² = 3⁶
Step 2: Evaluate if required: 3⁶ = 729.
Example 2 — Simplify x⁹/x⁴, where x ≠ 0.
Step 1: The bases are the same, so subtract the indices.
x⁹/x⁴ = x⁹⁻⁴
Example 3 — Simplify (a⁷b³)/(a²b), where a ≠ 0 and b ≠ 0.
Subtract indices for each matching base:
a⁷/a² = a⁵ and b³/b = b².
4. Zero and negative indices
A zero index gives 1 for any non-zero base. A negative index means take the reciprocal of the corresponding positive power.
a⁰ = 1 (a ≠ 0) and a⁻ⁿ = 1/aⁿ (a ≠ 0)
Example 4 — Simplify 5⁰ + 2⁻³.
5⁰ = 1 and 2⁻³ = 1/2³ = 1/8.
Therefore, 1 + 1/8 = 9/8.
Example 5 — Simplify (2a³b⁻²)².
Apply the power to every factor:
(2a³b⁻²)² = 2²a⁶b⁻⁴ = 4a⁶b⁻⁴.
Write the negative power as a reciprocal.
| Common mistake: 2⁻³ is not −2³. It equals 1/2³ = 1/8. |
5. Fractional indices
A fractional index connects powers with roots. The denominator gives the root and the numerator gives the power.
a¹/ⁿ = ⁿ√a and aᵐ/ⁿ = (ⁿ√a)ᵐ = ⁿ√(aᵐ)
When a is non-negative and the roots are real, either order of taking the root and then the power, or the power and then the root, gives the same result.
Example 6 — Evaluate 81³/⁴.
The denominator 4 indicates a fourth root; the numerator 3 indicates a cube.
81³/⁴ = (⁴√81)³ = 3³ = 27.
Example 7 — Evaluate 32²/⁵.
32²/⁵ = (⁵√32)².
Since ⁵√32 = 2, the expression is 2².
Example 8 — Simplify x¹/² × x³/², where x > 0.
Add the indices because the base is the same:
x¹/² × x³/² = x⁽¹/² + ³/²⁾ = x⁴/² = x².
6. Simplifying algebraic expressions with indices
Example 9 — Simplify (3x²y⁻¹)².
Square each factor:
(3x²y⁻¹)² = 3²x⁴y⁻² = 9x⁴y⁻².
Write with positive indices:
Example 10 — Simplify (a⁵b⁻²)(a⁻³b⁴), where a ≠ 0 and b ≠ 0.
Add indices for matching bases:
a⁵a⁻³ = a² and b⁻²b⁴ = b².
Example 11 — Simplify (16x⁸)¹/⁴, where x ≥ 0.
Apply the fourth root to each factor:
(16x⁸)¹/⁴ = 16¹/⁴ × x⁸/⁴ = 2x².
7. Solving basic exponential equations
When both sides of an exponential equation can be written with the same positive base (not equal to 1), equal powers imply equal indices.
Example 12 — Solve 2ˣ⁺¹ = 16.
Write 16 as a power of 2: 16 = 2⁴.
Then 2ˣ⁺¹ = 2⁴, so x + 1 = 4.
Example 13 — Solve 3²ˣ⁻¹ = 1/27.
Write 1/27 as a power of 3: 1/27 = 3⁻³.
Equate indices: 2x − 1 = −3.
Thus 2x = −2.
| Method: express both sides using the same base, equate the indices, solve the resulting equation, and check the result in the original equation. |
8. Common errors and examination tips
When multiplying powers with the same base, add indices; do not multiply them.
When dividing powers with the same base, subtract the denominator's index from the numerator's index.
A negative index indicates a reciprocal, not a negative value.
Use brackets carefully: (a²)³ = a⁶, whereas a² × a³ = a⁵.
State restrictions when variables appear in denominators.
For fractional indices, check that the root is defined in the real numbers.
In exponential equations, make sure the bases are identical before equating indices.
9. Practice exercise
Try the questions without looking at the answers. Show enough working for another student or teacher to follow your method.
A. Core skills
1. Simplify 5³ × 5⁴.
2. Simplify x⁸/x³, where x ≠ 0.
3. Simplify (m⁴)³.
4. Evaluate 7⁰ + 2³.
5. Express 4⁻² using a positive index.
6. Simplify (3a²b⁻¹)², where b ≠ 0.
B. Fractional indices
7. Evaluate 64²/³.
8. Evaluate 125²/³.
9. Simplify x¹/³ × x²/³, where x > 0.
10. Simplify (81y⁴)¹/², where y ≥ 0.
C. Equations and challenge
11. Solve 3ˣ⁻¹ = 27.
12. Solve 2²ˣ = 32.
13. Simplify (a⁶b⁻³)/(a²b⁻⁵), where a ≠ 0 and b ≠ 0.
14. Solve 5ˣ⁺² = 1/125.
15. Simplify (16x⁸)³/⁴, where x ≥ 0.
10. Answers to the practice exercise
1. 5⁷ = 78,125.
2. x⁵.
3. m¹².
4. 1 + 8 = 9.
5. 1/4² = 1/16.
6. 9a⁴/b².
7. (∛64)² = 4² = 16.
8. (∛125)² = 5² = 25.
9. x¹ = x.
10. 9y².
11. x − 1 = 3, so x = 4.
12. 2x = 5, so x = 5/2.
13. a⁴b².
14. x + 2 = −3, so x = −5.
15. (16x⁸)³/⁴ = 8x⁶.
For questions involving variables, the stated restrictions are part of the question. Where a question specifies a non-negative variable, the principal real root is intended.
Revision checklist and next part
☐ I can state and apply the laws of indices.
☐ I can rewrite negative indices as reciprocals.
☐ I can evaluate and simplify fractional powers.
☐ I can simplify algebraic expressions with indices.
☐ I can solve exponential equations by using a common base.
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