Powers Roots And Standard Form

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टीचर्स के लिए: Powers Roots And Standard Form (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

Powers & Roots

Powers (indices) show repeated multiplication: e.g., 6³ = 6 × 6 × 6.
Any non-zero number to the power of 0 equals 1: e.g., 3⁰ = 1.
Any number to the power of 1 equals itself: e.g., 3¹ = 3.
Square roots are the reverse of squaring; every positive number has two square roots (positive and negative).
The symbol √ denotes the positive square root only: e.g., √25 = 5.
Cube roots are the reverse of cubing; each number has only one cube root: e.g., ³√125 = 5.
For nth roots, if n is even there are two roots (positive and negative); if n is odd there is one.
The reciprocal of a number is 1 divided by that number; a number and its reciprocal multiply to 1.

Laws of Indices

a¹ = a – anything to the power 1 is itself.
a⁰ = 1 – anything (non-zero) to the power 0 is 1.
aᵐ × aⁿ = aᵐ⁺ⁿ – to multiply indices with the same base, add the powers.
aᵐ ÷ aⁿ = aᵐ⁻ⁿ – to divide indices with the same base, subtract the powers.
(aᵐ)ⁿ = aᵐⁿ – to raise a power to another power, multiply the powers.
(ab)ⁿ = aⁿbⁿ – to raise a product to a power, apply the power to each factor.
(a/b)ⁿ = aⁿ / bⁿ – to raise a fraction to a power, apply the power to numerator and denominator.
a⁻ⁿ = 1/aⁿ – a negative power means the reciprocal of the positive power.

Converting to & from Standard Form

Standard form is a × 10ⁿ where 1 ≤ a < 10 and n is an integer.
For large numbers (≥10), n is positive: count how many places the decimal moves left.
For small numbers (0 < number < 1), n is negative: count how many places the decimal moves right.
Example: 32 400 = 3.24 × 10⁴ (decimal moves 4 places left).
Example: 0.0000324 = 3.24 × 10⁻⁵ (decimal moves 5 places right).
To convert from standard form to ordinary number, move the decimal point n places (right if n>0, left if n<0).

Converting 32,400 to standard form

Operations with Standard Form

Use your calculator for calculations with standard form (calculator paper only).
Enter numbers using the ×10ˣ button and brackets: e.g., (3×10⁸)×(2×10⁻³).
If the calculator output is not in standard form, rewrite it: e.g., 37 500 000 = 3.75×10⁷.
Alternatively, rewrite a in standard form and apply index laws: e.g., 243×10²⁰ = (2.43×10²)×10²⁰ = 2.43×10²².

Worked Examples from Past Papers

Write 0.0018 in standard form: 1.8 × 10⁻³.
Write 6.09×10⁸ as an ordinary number: 609 000 000.
Find 5⁰: 1; find 5⁻²: 1/25.
Calculate (4.1×10⁻³)×(8.9×10⁷) = 3.649×10⁵ (using calculator).
Solve 5ˣ × 5³ = 5¹² → x = 9 (since x+3=12).
Solve 5ᵖ ÷ 5⁸ = 5¹³ → p = 21 (since p-8=13).

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साइन अप करें — फ्री लॉग इन

प्रैक्टिस सवाल

फ्री प्रीव्यू — 61 में से 8 सवाल। सभी देखने के लिए साइन अप करें।

1.Write down the value of 19⁰.
Easy
- A0
- B1
- C19
- D19⁰
2.The cost of building a ship was $153000000. Write 153000000 in standard form.
Easy
- A1.53 × 10⁸
- B1.53 × 10⁷
- C15.3 × 10⁷
- D1.53 × 10⁹
3.Write the number 40 in standard form.
Easy
- A4 × 10¹
- B40 × 10⁰
- C4.0 × 10¹
- D0.4 × 10²
4.Write 2020 in standard form.
Easy
- A2.02 × 10³
- B2.02 × 10⁴
- C20.2 × 10²
5.Write 15060 in standard form.
Easy
- A1.506 × 10⁴
- B1.506 × 10³
- C15.06 × 10³
- D1.506 × 10⁵
6.Write 72000 in standard form.
Easy
- A7.2 × 10⁴
- B7.2 × 10³
- C72 × 10³
- D7.2 × 10⁵
7.Write 0.0018 in standard form.
Easy
- A1.8 × 10^-3
- B1.8 × 10^-4
- C18 × 10^-4
- D1.8 × 10³
8.Write down the value of 25⁰.
Easy
- A0
- B1
- C25
- D25⁰

और सवाल

9.Write 640000 in standard form.
Easy
- A6.4 × 10⁵
- B6.4 × 10⁴
- C64 × 10⁴
- D6.4 × 10⁶
10.Write 0.0006 in standard form.
Easy
- A6 × 10^-4
- B6 × 10^-3
- C0.6 × 10^-3
- D6 × 10⁴
11.Write 0.00527 in standard form.
Easy
- A5.27 × 10^-3
- B5.27 × 10^-2
- C5.27 × 10^-4
- D0.527 × 10^-2
12.Write 2760000 in standard form.
Easy
- A2.76 × 10⁶
- B2.76 × 10⁷
- C27.6 × 10⁵
- D2.76 × 10⁵
13.Write 0.0000523 in standard form.
Easy
- A5.23 × 10^-5
- B5.23 × 10^-4
- C5.23 × 10^-6
- D0.523 × 10^-4
14.Write 19500000 in standard form.
Easy
- A1.95 × 10⁷
- B1.95 × 10⁶
- C19.5 × 10⁶
- D1.95 × 10⁸
15.Write 6.09 × 10⁸ as an ordinary number.
Easy
- A609000000
- B60900000
- C6090000000
16.Write 9.314 × 10⁵ as an ordinary number.
Easy
- A931400
- B93140
- C9314000
- D93140000
17.Write down the value of 3⁰.
Easy
- A0
- B1
- C3
- D30
18.Write 0.0605 in standard form.
Easy
- A6.05 × 10^-2
- B6.05 × 10^-3
- C6.05 × 10²
- D6.05 × 10³
19.Calculate (4.1 × 10^-3) × (8.9 × 10⁷). Give your answer in standard form.
Medium
- A3.649 × 10⁵
- B3.649 × 10⁴
- C3.649 × 10⁶
- D3.649 × 10^-4
20.Calculate (1.63 × 10¹²) × (2.47 × 10^-1). Give your answer in standard form.
Medium
- A4.0261 × 10¹¹
- B4.0261 × 10¹²
- C4.0261 × 10¹⁰
- D4.0261 × 10¹³
21.Write 2.06 × 10^-2 as an ordinary number.
Easy
- A0.0206
- B0.206
- C206
- D0.00206
22.Find the value of 5^-2.
Easy
- A1/25
- B1/10
- C25
- D-25
23.Write down the reciprocal of 15.
Easy
- A1/15
- B15
- C-15
- D15/1
24.Write 0.015 in standard form.
Easy
- A1.5 × 10^-2
- B1.5 × 10^-3
- C1.5 × 10²
- D1.5 × 10³
25.Find the value of 6⁰ + 6².
Medium
- A37
- B36
- C1
- D12
26.Find the value of 5^-4.
Easy
- A1/625
- B1/20
- C625
- D-625
27.Calculate (6.3 × 10²) ÷ (7 × 10^-3). Write your answer in standard form.
Medium
- A9 × 10⁴
- B9 × 10⁵
- C9 × 10³
- D9 × 10⁶
28.Find the value of x when 5^x × 5³ = 5¹².
Easy
- A9
- B4
- C15
- D36
29.Find the value of p when 5^p ÷ 5⁸ = 5¹³.
Easy
- A21
- B5
- C104
30.Write 45000 in standard form.
Easy
- A4.5 × 10⁴
- B4.5 × 10⁵
- C45 × 10³
- D4.5 × 10³
31.Write 4.2 × 10^-3 as an ordinary number.
Easy
- A0.0042
- B0.042
- C4200
- D0.00042
32.Calculate (8.1 × 10³) + (7.9 × 10⁴). Give your answer in standard form.
Medium
- A8.71 × 10⁴
- B8.71 × 10³
- C8.71 × 10⁵
- D8.71 × 10²
33.The number 0.0000456 is written in standard form as 4.56 × 10ⁿ. What is the value of n?
Easy
- A-5
- B5
- C-4
- D4
34.The mass of a grain of sand is 4.5 × 10^-3 grams. What is this as an ordinary number?
Easy
- A0.0045
- B0.045
- C4500
- D0.00045
35.Calculate (3.2 × 10⁵) × (4.5 × 10^-2). Give your answer in standard form.
Medium
- A1.44 × 10⁴
- B1.44 × 10³
- C1.44 × 10⁵
- D1.44 × 10²
36.The distance from Earth to Mars is 2.25 × 10⁸ km. A spacecraft travels at 5.8 × 10⁴ km/h. How many hours does the journey take? Give your answer in standard form.
Hard
- A3.88 × 10³
- B3.88 × 10⁴
- C3.88 × 10²
- D3.88 × 10⁵
37.Find the value of 5⁰ + 5^-2.
Medium
- A1.04
- B1.25
- C0.04
- D1.2
38.Simplify 8⁴ ÷ 8^-2.
Medium
- A8⁶
- B8²
- C8⁸
- D8^-2
39.Mercury is 5.8 × 10⁷ km from the Sun. Jupiter is 7.22 × 10⁸ km further from the Sun than Mercury. Earth is 1.5 × 10⁸ km from the Sun. How many times further from the Sun is Jupiter than Earth?
Hard
- A5.2
- B4.8
- C5.0
- D5.4
40.Find the value of x such that 2^x × 2⁵ = 2¹².
Easy
- A7
- B17
- C60
- D2
41.What is the value of 7⁰?
Easy
- A0
- B1
- C7
- D70
42.Which of the following is the correct standard form for 0.00045?
Easy
- A4.5 × 10^-4
- B4.5 × 10⁴
- C45 × 10^-5
- D4.5 × 10^-3
43.The square root of a positive number is always positive.
Medium
True or false?
44.Simplify 3⁵ × 3².
Easy
- A3⁷
- B3¹⁰
- C9⁷
- D3³
45.Write 6.2 × 10⁵ as an ordinary number.
Easy
- A620000
- B62000
- C6200000
- D0.000062
46.Simplify 2⁸ ÷ 2^-3.
Medium
- A2¹¹
- B2⁵
- C2²⁴
- D2^-11
47.Which of the following are equal to 10^-3? (Select all that apply)
Easy
- A0.001
- B1/1000
- C1/100
- D0.0001
- E1/10000
48.Which of the following are equal to 64? (Select all that apply)
Medium
- A4³
- B2⁶
- C8²
- D(-8)²
- E(-4)³
49.Match each expression with its simplified form using the laws of indices.
Medium
- a⁵ × a³
- a⁷ ÷ a²
- (a⁴)²
- a⁰
- a⁸
- a⁵
- a⁸
- 1
50.Put these numbers in order from smallest to largest.
Medium
- 3.2 × 10^-4
- 0.00032
- 3.2 × 10^-5
- 0.000032
51.Write down the value of 2³.
Easy
52.Find the value of 4^-2.
Easy
53.Calculate (2.5 × 10³) × (4 × 10^-2). Give your answer in standard form.
Medium
54.Complete the sentence.
Easy
The reciprocal of 8 is ____.
55.Calculate (4.8 × 10⁵) ÷ (1.2 × 10^-2). Give your answer in standard form.
Medium
56.Complete the sentence using the words 'positive' or 'negative'.
Easy
The square root of a positive number has ____ and ____ roots.
57.Explain why the value of 5⁰ is 1.
Hard
58.Write down the value of 2³.
Easy
- A8
- B7
- C9
- D10
59.Find the value of 4^-2.
Easy
- A0.0625
- B1.0625
- C-0.9375
- D2.0625
60.Calculate (2.5 × 10³) × (4 × 10^-2). Give your answer in standard form.
Medium
- A100
- B120
- C90
- D110
61.Calculate (4.8 × 10⁵) ÷ (1.2 × 10^-2). Give your answer in standard form.
Medium
- A36000000
- B44000000
- C48000000
- D40000000

रिवीज़न नोट्स

## Powers, Roots & Standard Form ### Powers & Roots - **Powers (indices)**: small number showing repeated multiplication. e.g., 6³ = 6×6×6. - **Any non-zero number to power 0 = 1** (e.g., 3⁰ = 1). - **Any number to power 1 = itself** (e.g., 3¹ = 3). - **Square roots**: √25 = 5 (positive root). Every positive number has two square roots (±). Negative numbers have no real square root. - **Cube roots**: ³√125 = 5. Each number has one cube root. - **nth roots**: For even n, positive and negative roots exist; negative numbers have no real even root. For odd n, one real root. - **Reciprocal**: 1 divided by the number. e.g., reciprocal of 2 is ½. Written as index -1: 5⁻¹ = 1/5. ### Laws of Indices - a¹ = a - a⁰ = 1 - aᵐ × aⁿ = aᵐ⁺ⁿ - aᵐ ÷ aⁿ = aᵐ⁻ⁿ - (aᵐ)ⁿ = aᵐⁿ - (ab)ⁿ = aⁿbⁿ - (a/b)ⁿ = aⁿ/bⁿ - a⁻ⁿ = 1/aⁿ ### Standard Form - **Form**: A × 10ⁿ where 1 ≤ A < 10 and n is integer. - **Large numbers**: n positive. Move decimal left. e.g., 32400 = 3.24 × 10⁴. - **Small numbers**: n negative. Move decimal right. e.g., 0.0000324 = 3.24 × 10⁻⁵. - **Converting from standard form**: Move decimal n places (right if n>0, left if n<0). ### Operations with Standard Form - Use calculator for calculations (calculator paper). - Multiply/divide numbers and powers separately, then adjust to standard form. - Example: (1.275×10⁶) ÷ (3.4×10⁻²) = 3.75×10⁷.

Power (Index): A small number showing how many times a base is multiplied by itself.
Base: The big number that is raised to a power.
Square root: A value that when squared equals the original number. Notation: √.
Cube root: A value that when cubed equals the original number. Notation: ³√.
Reciprocal: 1 divided by a number. The reciprocal of x is 1/x.
Standard form: A number written as A × 10ⁿ where 1 ≤ A < 10 and n is an integer.
Laws of indices: Rules for simplifying expressions with powers, e.g., aᵐ × aⁿ = aᵐ⁺ⁿ.

फ्लैशकार्ड

What is the value of any non-zero number raised to the power of 0?

Any non-zero number to the power of 0 is equal to 1. For example, 3⁰ = 1.

What is the value of any number raised to the power of 1?

Any number to the power of 1 is equal to itself. For example, 3¹ = 3.

What are the two square roots of 25?

The two square roots of 25 are 5 and -5, because 5² = 25 and (-5)² = 25.

What does the notation √25 represent?

The notation √25 refers to the positive square root, which is 5.

What is the cube root of 125?

The cube root of 125 is 5, because 5³ = 125.

How many real cube roots does a number have?

Every positive and negative number has exactly one real cube root.

What is the reciprocal of a number?

The reciprocal of a number is 1 divided by that number. For example, the reciprocal of 2 is 1/2.

How do you write the reciprocal of 5 using an index?

The reciprocal of 5 is written as 5^-1.

What is the law of indices for multiplying powers with the same base?

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

What is the law of indices for dividing powers with the same base?

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

What is the law of indices for raising a power to another power?

To raise a power to another power, multiply the powers: (a^m)ⁿ = a^mn.

What does a negative index represent?

A negative index represents the reciprocal: a^-n = 1/(aⁿ). For example, 2^-4 = 1/16.

Write 153000000 in standard form.

153000000 = 1.53 × 10⁸.

Write 0.0018 in standard form.

0.0018 = 1.8 × 10^-3.

Write 6.09 × 10⁸ as an ordinary number.

6.09 × 10⁸ = 609000000.

Write 2.06 × 10^-2 as an ordinary number.

2.06 × 10^-2 = 0.0206.

Calculate (4.1 × 10^-3) × (8.9 × 10⁷) and give your answer in standard form.

4.1 × 8.9 = 36.49, and 10^-3 × 10⁷ = 10⁴, so the product is 36.49 × 10⁴ = 3.649 × 10⁵.

Calculate (1.63 × 10¹²) × (2.47 × 10^-1) and give your answer in standard form.

1.63 × 2.47 = 4.0261, and 10¹² × 10^-1 = 10¹¹, so the product is 4.0261 × 10¹¹.

Calculate (8.1 × 10³) + (7.9 × 10⁴) and give your answer in standard form.

Convert to same power: 8.1 × 10³ = 0.81 × 10⁴, then add: (0.81 + 7.9) × 10⁴ = 8.71 × 10⁴.

Calculate (6.3 × 10²) ÷ (7 × 10^-3) and give your answer in standard form.

6.3 ÷ 7 = 0.9, and 10² ÷ 10^-3 = 10⁵, so the result is 0.9 × 10⁵ = 9 × 10⁴.

Find the value of x when 5^x × 5³ = 5¹².

Using the law a^m × aⁿ = a^m+n, we have 5^x+3 = 5¹², so x+3 = 12, hence x = 9.

Find the value of p when 5^p ÷ 5⁸ = 5¹³.

Using the law a^m ÷ aⁿ = a^m-n, we have 5^p-8 = 5¹³, so p-8 = 13, hence p = 21.

Find the value of 2^-4 without a calculator.

2^-4 = 1/(2⁴) = 1/16.

Find the value of 5^-2.

5^-2 = 1/(5²) = 1/25.

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