Algebraic Roots And Indices

खेलकर सीखें

इन सवालों के जवाब देकर एनर्जी कमाएं, फिर मछली पकड़ें और घूमें। कोई अकाउंट नहीं चाहिए।

टीचर्स के लिए: Algebraic Roots And Indices (Maths [CIE], Core) के लिए इस्तेमाल के लिए तैयार लेसन स्लाइड्स, रिवीज़न नोट्स — इन्हें अपने लेसन में इस्तेमाल करें, या टॉपिक को एक इंटरैक्टिव क्लास एक्टिविटी की तरह चलाएं जिसे आपके स्टूडेंट्स लाइव गेम की तरह खेलें।

लेसन नोट्स

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.

स्लाइड्स

Step through every slide for this topic — plus flashcards and revision notes — with a free account.

साइन अप करें — फ्री लॉग इन

प्रैक्टिस सवाल

फ्री प्रीव्यू — 53 में से 8 सवाल। सभी देखने के लिए साइन अप करें।

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

और सवाल

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

रिवीज़न नोट्स

## 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.

फ्लैशकार्ड

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.

Unlock all 53 questions, flashcards & more

इस टॉपिक के हर सवाल, स्लाइड्स, फ्लैशकार्ड और रिवीज़न नोट्स देखने के लिए फ्री अकाउंट बनाएं।

साइन अप करें — फ्री लॉग इन

पास्ट पेपर

इस टॉपिक के लिए पास्ट-पेपर प्रैक्टिस जल्द आ रही है।

जल्द आ रहा है