Algebraic Roots And Indices

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Lesson notes

Laws of Indices

Index laws apply to both numbers and algebra.
a¹ = a: any term to the power 1 is itself.
a⁰ = 1: any non‑zero term to the power 0 equals 1.
aᵐ × aⁿ = aᵐ⁺ⁿ: add powers when multiplying same base.
aᵐ ÷ aⁿ = aᵐ⁻ⁿ: subtract powers when dividing same base.
(aᵐ)ⁿ = aᵐⁿ: multiply powers when raising a power to another power.
(ab)ⁿ = aⁿbⁿ: apply the power to each factor in a product.
(a/b)ⁿ = aⁿ / bⁿ: apply the power to numerator and denominator.

Negative and Zero Indices

a⁻¹ = 1/a: a negative index gives the reciprocal.
a⁻ⁿ = 1/aⁿ: general rule for negative powers.
a⁰ = 1 for any a ≠ 0.
Example: 2⁻³ = 1/2³ = 1/8.

Simplifying Expressions with Indices

Work out the number part and algebra part separately.
Multiply coefficients and use index laws for variables.
Example: (3x⁷) × (6x⁴) = 18x¹¹.
Example: 6x⁷ ÷ 3x⁴ = 2x³.
Example: (3x⁷)² = 9x¹⁴.

Solving Equations with Unknown Indices

Write both sides with the same base.
Then equate the indices to find the unknown.
Example: 5ˣ = 125 → 5ˣ = 5³ → x = 3.
Example: 2ˣ = 1/8 → 2ˣ = 2⁻³ → x = –3.

Worked Examples from Syllabus

Simplify (u⁵)⁵ = u²⁵ using (aᵐ)ⁿ = aᵐⁿ.
If qˣ = (q² × q⁵)/q¹⁰, simplify numerator: q²⁺⁵ = q⁷.
Then q⁷/q¹⁰ = q⁷⁻¹⁰ = q⁻³, so x = –3.

Common Exam Question Types

Find the value of x in 6¹⁰ × 6ˣ = 6² → x = –8.
Find x in 5ˣ × 5³ = 5¹² → x = 9.
Simplify p² × p⁴ = p⁶.
Simplify m¹⁵ ÷ m⁵ = m¹⁰.
Simplify (k³)⁵ = k¹⁵.

Medium and Hard Questions

Simplify 5 × x⁰ = 5 × 1 = 5.
Simplify 2x³ × 3x² = 6x⁵.
Simplify 4p⁵q³ × p²q⁻⁴ = 4p⁷q⁻¹.
Simplify 8t⁸ ÷ 4t⁴ = 2t⁴.
Solve 4ʷ = 1/16 → 4ʷ = 4⁻² → w = –2.

Slides

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Practice questions

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1.What is the value of any non-zero number raised to the power of 0?
Easy
- A0
- B1
- Cthe number itself
- Dundefined
2.Simplify p² × p⁴.
Easy
- Ap⁶
- Bp⁸
- C2p⁶
- Dp²
3.Simplify m¹⁵ ÷ m⁵.
Easy
- Am¹⁰
- Bm²⁰
- Cm³
- Dm⁷⁵
4.Simplify (k³)⁵.
Easy
- Ak⁸
- Bk¹⁵
- Ck²
- Dk¹²⁵
5.If 9¹² ÷ 9^w = 9⁴, find the value of w.
Medium
- A8
- B16
- C3
- D48
6.Simplify (w⁵)⁴.
Easy
- Aw⁹
- Bw²⁰
- Cw⁶²⁵
- Dw¹
7.Simplify t²¹ ÷ t⁷.
Easy
- At¹⁴
- Bt²⁸
- Ct³
- Dt¹⁴⁷
8.Simplify (u⁵)⁵.
Easy
- Au¹⁰
- Bu²⁵
- Cu³¹²⁵
- Du⁰

Revision notes

## Algebraic Roots & Indices ### Laws of Indices (same base) - **Multiplication**: \(a^m \times a^n = a^{m+n}\) Example: \(p^2 \times p^4 = p^6\) - **Division**: \(a^m \div a^n = a^{m-n}\) Example: \(m^{15} \div m^5 = m^{10}\) - **Power of a power**: \((a^m)^n = a^{mn}\) Example: \((k^3)^5 = k^{15}\) - **Power of a product**: \((ab)^n = a^n b^n\) - **Power of a fraction**: \(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\) ### Special Indices - \(a^1 = a\) - \(a^0 = 1\) - \(a^{-1} = \frac{1}{a}\), \(a^{-n} = \frac{1}{a^n}\) (negative power = reciprocal) ### Simplifying Expressions - Separate number and algebra parts: \(3x^7 \times 6x^4 = (3\times6)(x^{7+4}) = 18x^{11}\) \(\frac{6x^7}{3x^4} = \frac{6}{3}x^{7-4} = 2x^3\) \((3x^7)^2 = 3^2 x^{14} = 9x^{14}\) ### Solving Equations with Unknown Index - Write both sides with the same base, then equate indices. Example: \(5^x = 125 \Rightarrow 5^x = 5^3 \Rightarrow x=3\) Example: \(2^x = \frac{1}{8} \Rightarrow 2^x = 2^{-3} \Rightarrow x=-3\) ### Worked Example Simplify \((u^5)^5\): \((u^5)^5 = u^{5\times5} = u^{25}\) If \(q^x = \frac{q^2 \times q^5}{q^{10}}\), find \(x\): Numerator: \(q^2 \times q^5 = q^7\) Division: \(\frac{q^7}{q^{10}} = q^{-3}\) So \(q^x = q^{-3} \Rightarrow x = -3\)

Base: The number or variable that is raised to a power (e.g., in \(a^n\), \(a\) is the base).
Index (plural: indices): The power or exponent to which a base is raised (e.g., in \(a^n\), \(n\) is the index).
Law of indices: Rules for simplifying expressions involving powers: multiplication (add indices), division (subtract indices), power of a power (multiply indices).
Reciprocal: The multiplicative inverse; \(a^{-1} = \frac{1}{a}\).
Simplify: To combine terms using index laws to write an expression in a more compact form.

Flashcards

What is the value of any number (or variable) raised to the power of 0?

Any number (or variable) raised to the power of 0 equals 1. For example, b⁰ = 1.

What is the value of any number (or variable) raised to the power of 1?

Any number (or variable) raised to the power of 1 is itself. For example, x¹ = x.

Law of indices: a^m × aⁿ = ?

a^m × aⁿ = a^m+n. To multiply indices with the same base, add their powers.

Law of indices: a^m ÷ aⁿ = ?

a^m ÷ aⁿ = a^m-n. To divide indices with the same base, subtract their powers.

Law of indices: (a^m)ⁿ = ?

(a^m)ⁿ = a^m×n. To raise an index to a new power, multiply the powers.

Law of indices: (ab)ⁿ = ?

(ab)ⁿ = aⁿ × bⁿ. To raise a product to a power, apply the power to each factor.

Law of indices: (a/b)ⁿ = ?

(a/b)ⁿ = aⁿ / bⁿ. To raise a fraction to a power, apply the power to numerator and denominator.

What does a negative index represent?

A negative index represents the reciprocal. a^-n = 1/(aⁿ). For example, k^-3 = 1/k³.

Simplify: p² × p⁴

p² × p⁴ = p²⁺⁴ = p⁶.

Simplify: m¹⁵ ÷ m⁵

m¹⁵ ÷ m⁵ = m^15-5 = m¹⁰.

Simplify: (k³)⁵

(k³)⁵ = k^3×5 = k¹⁵.

Simplify: (w⁵)⁴

(w⁵)⁴ = w^5×4 = w²⁰.

Simplify: t²¹ ÷ t⁷

t²¹ ÷ t⁷ = t^21-7 = t¹⁴.

Simplify: (u⁵)⁵

(u⁵)⁵ = u^5×5 = u²⁵.

Simplify: (x⁸)³

(x⁸)³ = x^8×3 = x²⁴.

Simplify: (y⁵)³

(y⁵)³ = y^5×3 = y¹⁵.

Simplify: (x⁵)²

(x⁵)² = x^5×2 = x¹⁰.

If x³ × x⁶ = x^m, find m.

x³ × x⁶ = x³⁺⁶ = x⁹, so m = 9.

If (x²)ⁿ = x¹², find n.

(x²)ⁿ = x²ⁿ = x¹², so 2n = 12, hence n = 6.

Solve: 5^x = 125

125 = 5³, so 5^x = 5³, therefore x = 3.

Solve: 2^x = 1/8

1/8 = 1/2³ = 2^-3, so 2^x = 2^-3, therefore x = -3.

Simplify: 5 × x⁰

x⁰ = 1, so 5 × 1 = 5.

Simplify: 2x³ × 3x²

Multiply coefficients: 2×3=6. Add powers: x³⁺²=x⁵. Result: 6x⁵.

Simplify: 5m² × 2m³

5×2=10, m²⁺³=m⁵, so 10m⁵.

Simplify: 4p⁵ q³ × p² q^-4

4 × 1 = 4; p⁵⁺²=p⁷; q^3+(-4)=q^-1=1/q. Result: 4p⁷/q.

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