Algebraic Roots And Indices

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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.

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Câu hỏi luyện tập

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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

Thêm câu hỏi

9.Simplify (x⁸)³.
Easy
- Ax¹¹
- Bx²⁴
- Cx⁵¹²
- Dx⁵
10.Simplify (y⁵)³.
Easy
- Ay⁸
- By¹⁵
- Cy¹²⁵
- Dy²
11.Simplify 5 × x⁰.
Easy
- A5
- B5x
- C1
- D0
12.Simplify 2x³ × 3x².
Medium
- A6x⁵
- B6x⁶
- C5x⁵
- D6x¹
13.Simplify 5m² × 2m³.
Medium
- A10m⁵
- B10m⁶
- C7m⁵
- D10m¹
14.If x¹⁰ ÷ x^k = x³, find k.
Medium
- A7
- B13
- C30
- D3
15.If y¹⁰ × yⁿ = 1, find n.
Medium
- A-10
- B10
- C0
- D1
16.Simplify 4p⁵ q³ × p² q^-4.
Medium
- A4p⁷ q^-1
- B4p¹⁰ q^-12
- C4p⁷ q⁷
- D4p³ q^-1
17.Simplify \frac{6x⁷}{3x⁴}.
Medium
- A2x³
- B2x¹¹
- C18x³
- D18x¹¹
18.Simplify (3x⁷)².
Medium
- A9x¹⁴
- B9x⁹
- C6x¹⁴
- D3x¹⁴
19.If q^x = \frac{q² \times q⁵}{q¹⁰}, find the value of x.
Medium
- A-3
- B3
- C7
- D-7
20.Simplify 8t⁸ \div 4t⁴.
Medium
- A2t⁴
- B2t²
- C4t⁴
- D12t⁴
21.Simplify \frac{x³ y⁴}{x⁵ y³}.
Medium
- Ax^-2 y
- Bx² y
- Cx^-2 y⁷
- Dx⁸ y⁷
22.Simplify w⁷ \div w^-2.
Medium
- Aw⁹
- Bw⁵
- Cw¹⁴
- Dw^-9
23.If 4^w = \frac{1}{16}, find the value of w.
Medium
- A-2
- B2
- C4
- D-4
24.Simplify x³ \times x⁶.
Easy
- Ax⁹
- Bx¹⁸
- Cx³
- Dx²
25.If (x²)ⁿ = x¹², find the value of n.
Medium
- A6
- B10
- C14
- D24
26.If 2⁸ \times 2^m = 2⁶, work out the value of m.
Easy
- A-2
- B2
- C14
- D-14
27.If 5ⁿ \div 5⁴ = 5⁶, work out the value of n.
Easy
- A10
- B2
- C-2
- D24
28.Simplify (x⁵)².
Easy
- Ax¹⁰
- Bx⁷
- Cx²⁵
- Dx³
29.Find the value of x when 5^x \times 5³ = 5¹².
Easy
- A9
- B15
- C4
- D36
30.Simplify 3x² \times 2x³.
Medium
- A6x⁵
- B6x⁶
- C5x⁵
- D5x⁶
31.What is the value of 5 \times x⁰ when x is not zero?
Easy
- A5
- B1
- C0
- D5x
32.Simplify \frac{4p⁵ q³}{p² q^-4}.
Medium
- A4p³ q⁷
- B4p⁷ q^-1
- C4p³ q^-1
- D4p⁷ q⁷
33.Simplify x¹.
Easy
- Ax
- B1
- C0
- Dx⁰
34.The expression a⁰ is always equal to 1.
Medium
True or false?
35.Match each expression to its simplified form.
Medium
- x³ × x²
- x⁵ ÷ x²
- (x³)²
- x⁵
- x³
- x⁶
36.Arrange these values in increasing order: 2⁰, 2^-1, 2¹, 2².
Medium
- 2⁰
- 2^-1
- 2¹
- 2²
37.Simplify 3x⁰.
Easy
- A3
- B0
- C1
- D3x
38.Which of the following are equal to 1? (Select all that apply)
Medium
- A5⁰
- B0⁵
- C1⁵
- D(-3)⁰
- E0⁰
39.Match each expression in the left column to its simplified form in the right column.
Medium
- x² \times x³
- (x²)³
- x⁵ \div x²
- x⁰
- x⁵
- x⁶
- x³
- 1
40.Arrange these values in increasing order (smallest first).
Medium
- 2^-3
- 2⁰
- 2^-1
- 2²
41.Which of the following are equivalent to 1? (Select all that apply.)
Medium
- A5⁰
- B0⁵
- C1¹⁰
- D(-3)⁰
- E0⁰
42.The expression a^-n is equal to 1/(aⁿ) for any non-zero a.
Easy
True or false?
43.Evaluate 6⁰.
Easy
44.Fill in the blank to complete the law of indices.
Easy
a^m × aⁿ = a^____
45.Find the value of n such that 3ⁿ = 1/9.
Medium
46.Write 5^-2 as a fraction in its simplest form.
Easy
47.If 2^x = 1/32, find x.
Medium
48.Complete the law: a^-n = ___
Easy
a^-n = ____
49.Find the value of x such that 4^x = 1/64.
Medium
50.Evaluate 6⁰.
Easy
- A2
- B0
- C3
- D1
51.Find the value of n such that 3ⁿ = 1/9.
Easy
- A-1
- B-3
- C0
- D-2
52.If 2^x = 1/32, find x.
Easy
- A-4
- B-5
- C-6
- D-3
53.Find the value of x such that 4^x = 1/64.
Easy
- A-3
- B-1
- C-4
- D-2

Ghi chú ôn tập

## 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.

Thẻ ghi nhớ

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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