Indices – SS1 Mathematics Lesson Note (First Term)

Indices (or powers) show how many times a base is multiplied by itself, as in 2³ = 2 × 2 × 2 = 8.

Behavioural objectives

By the end of the lesson, students should be able to:

1. state and use the laws of indices to simplify expressions;
2. simplify expressions in the form (x^a)^b;
3. explain why x^1/2 = √x and x^1/3 = ∛x;
4. simplify expressions with fractional and negative indices;
5. solve equations that involve indices.

What are indices?

An index is the small raised number that tells how many times the base is multiplied by itself.

• Base: the number or letter being multiplied, such as 2 in 2³.
• Index (power or exponent): the raised number, such as 3 in 2³.
• Plural: the plural of index is indices.
• Zero index: any non-zero base with index 0 gives 1.
• Negative index: gives the reciprocal of the power.
• Fractional index: stands for a root, so x^1/2 means the square root of x.

What are the laws of indices?

The laws of indices are rules for multiplying, dividing and reducing powers that have the same base.

1. x^a × x^b = x^a+b
2. x^a ÷ x^b = x^a-b
3. x⁰ = 1 (x is not zero)
4. x^-a = 1/x^a

How do the four laws work?

Worked examples

Simplify:

1. 10⁵ × 10⁴
   = 10⁵⁺⁴
   = 10⁹
2. a³ × a⁴
   = a³⁺⁴
   = a⁷
3. m⁸ ÷ m⁵
   = m^8-5
   = m³
4. 24x⁶ ÷ 8x⁴
   = (24/8) × x^6-4
   = 3x²
5. 19⁸ ÷ 19⁸
   = 19^8-8
   = 19⁰
   = 1

Classwork 1

Simplify:

1. 6 × Z⁰
2. 4^-3
3. r × r × r × r^-5

How do you find the power of a power?

To find the power of a power, multiply the indices: (x^a)^b = x^a×b = x^ab.

• Brackets matter: the index outside applies to every factor inside, including numbers and signs.
• Even power of a negative: (-a)⁴ is positive, because the four negative signs pair up.
• Negative index: change to a reciprocal before expanding.

Worked examples

Simplify:

1. (x²)³
   = x^2×3
   = x⁶
2. (y⁴)²
   = y^4×2
   = y⁸
3. (3^-2)^-3
   = 3^(-2)×(-3)
   = 3⁶
   = 3 × 3 × 3 × 3 × 3 × 3
   = 27 × 27
   = 729
4. (-3d³)²
   = (-3)² × (d³)²
   = (-3) × (-3) × d⁶
   = 9d⁶
5. a⁶(-a)^-4
   = a⁶ × 1/(-a)⁴
   = a⁶ / [(-a) × (-a) × (-a) × (-a)]
   = a⁶/a⁴
   = a^6-4
   = a²

Classwork 2

Simplify:

1. (h⁴)^-5
2. (-4u²v)³
3. (-x³)² ÷ x⁴
4. -(d²) ÷ d⁴ × -d
5. (-c)² × (c)⁴ ÷ (-c³)

What are fractional indices?

Fractional indices are indices in the form x^1/a or x^a/b, and they are another way of writing roots.

Why is the square root x^1/2?

√x is the square root of x, and √x × √x = x.

1. Let √x = x^p.
2. Then x^p × x^p = √x × √x = x¹.
3. So x^2p = x¹.
4. Equate the indices: 2p = 1, so p = ½.
5. Thus √x = x^1/2.

Why is the cube root x^1/3?

∛x is the cube root of x, and ∛x × ∛x × ∛x = x.

• Example 1: ∛8 = 2, since 2 × 2 × 2 = 8.
• Example 2: ∛-27 = -3, since (-3) × (-3) × (-3) = -27.

1. Let ∛x = x^q.
2. Then x^q × x^q × x^q = x¹, so x^3q = x¹.
3. Equate the powers: 3q = 1, so q = ⅓.
4. Thus ∛x = x^1/3.

What are the general rules for fractional indices?

• Unit fraction: x^1/a = ^a√x, the a-th root of x.
• Other fractions: x^a/b = ^b√(x^a) or (^b√x)^a.
• Tip: take the root first, then the power, because the numbers stay smaller.

For example, x^2/3 can be worked out in two ways:

1. x^2/3 = x^{2 × 1/3} = (x²)^1/3 = ∛(x²)
2. x^2/3 = x^{1/3 × 2} = (x^1/3)² = (∛x)²

Worked examples

Simplify:

1. 8^-2/3
   = 1/8^2/3
   = 1/(∛8)²
   = 1/2²
   = 1/4
2. 4^1/6 × 4^1/3
   = 4^{1/6 + 1/3}
   = 4^3/6
   = 4^1/2
   = √4
   = 2
3. (16/81)^-3/4
   = 1/(16/81)^3/4
   = 1/(⁴√(16/81))³
   = 1/(2/3)³
   = 1 ÷ (8/27)
   = 1 × (27/8)
   = 27/8
4. √(72a³b^-2 / 2a⁵b^-6)
   = (72a³b^-2 / 2a⁵b^-6)^1/2
   = ((72/2) × a^3-5 × b^-2-(-6))^1/2
   = √(36a^-2 × b^-2+6)
   = √(36a^-2 × b⁴)
   = √36 × (a^-2 × b⁴)^1/2
   = 6 × a^{-2 × 1/2} × b^{4 × 1/2}
   = 6a^-1 × b²
   = 6 × (1/a) × b²
   = 6b²/a

Classwork 3

Simplify:

1. (125)^-1/3
2. (18/32)^-3/2
3. (∛4)^1.5
4. 64^-5/6
5. √(1 9/16)

How do you solve equations with indices?

To solve an equation with indices, simplify with the laws of indices, then equate the powers or raise both sides to a suitable power.

• Same base on both sides: write both sides with one base, then equate the powers.
• Unknown inside a power: isolate the power, then raise both sides to the reciprocal of its index.
• Negative index: rewrite as a fraction first.

Worked examples

Solve the following equations:

1. 2r^-3 = -16

1. Divide both sides by 2: r^-3 = -8.
2. Write as a fraction: 1/r³ = -8/1.
3. Cross multiply: -8r³ = 1.
4. Divide by -8: r³ = -1/8.
5. Take the cube root of both sides: ∛(r³) = ∛(-1/8).
6. So r = -1/2.

2. 5x = 40x^-1/2

1. Divide both sides by 5: x = 8x^-1/2.
2. Write the negative index as a fraction: x = 8 × 1/x^1/2.
3. Cross multiply: x × x^1/2 = 8.
4. Add the indices: x^{1 + 1/2} = 8, so x^3/2 = 8.
5. Raise both sides to the power 2/3: (x^3/2)^2/3 = 8^2/3.
6. Simplify: x¹ = (∛8)².
7. So x = 2² = 4.

3. 4^c-1 = 64

1. Change both sides to the same base: 4^c-1 = 4³.
2. Equate the powers: c – 1 = 3.
3. Add 1 to both sides: c = 3 + 1.
4. So c = 4.

Classwork 4

Solve the following equations:

1. a^2/3 = 9
2. 2x³ = 54

Evaluation

1. State the four laws of indices given in this lesson.
2. Write x^1/2 and x^1/3 in root form.
3. Write x^a/b in root form in two ways.
4. Simplify (h⁴)^-5.
5. Solve a^2/3 = 9.

Assignment

Answer the following questions.

1. Simplify 3³ × 6^-3 × 2⁵.
   (a) 0 (b) 1 (c) 2 (d) 4 (e) 12
2. Calculate the value of (27/125)^1/3 × (4/9)^1/2.
   (a) 12/25 (b) 2/5 (c) 3/5 (d) 9/10 (e) 10/9
3. If 5p^-3 = 8 × 5^-2, find the value of p.
   (a) 8/125 (b) 2/5 (c) 4/5 (d) 8/5 (e) 5/2
4. If x² = 81^1½, x = ……..
   (a) 3 (b) 9 (c) 18 (d) 27 (e) 54
5. Simplify (x^1⅓)³ × 1/x⁴.
   (a) 1 (b) x¹ (c) x³ (d) x⁰ (e) 2
6. Evaluate 9^1/2 × 27^2/3.
7. Solve the following equations:
   1. y^-2 = 9
   2. (2s)^1/2 = 9
   3. 2^n-1 = 16