Laws of Indices and Square Numbers

 Laws of Indices and Square Numbers

Laws of Indices

Indices are used to show that a number is being multiplied by itself. For example:
  • 2³ means 2 × 2 × 2
  • 3² means 3 × 3
The laws of indices are used to simplify expressions that involve indices.

Positive Indices

A positive index shows that a number is being multiplied by itself. For example:
  • 2³ = 2 × 2 × 2 = 8
  • 3² = 3 × 3 = 9

Zero Indices

Any number with a zero index is equal to 1. For example:
  • 2⁰ = 1
  • 3⁰ = 1

Laws of Indices

There are several laws of indices that can be used to simplify expressions:
  1. Multiplication Law: aᵐ × aⁿ = aᵐ⁺ⁿ
  2. Division Law: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
  3. Power Law: (aᵐ)ⁿ = aᵐ×ⁿ

Applying the Laws of Indices

Using the laws of indices, we can simplify expressions such as:
  • 2³ × 2² = 2³⁺² = 2⁵
  • 3² ÷ 3¹ = 3²⁻¹ = 3¹
  • (2²)³ = 2²×³ = 2⁶

  • Here is the revised text:

    Laws of Indices

    Multiplication Law

    When multiplying powers with the same base, add the exponents.
    aᵐ × aⁿ = aᵐ⁺ⁿ

    Examples

    1. 2³ × 2⁵ = 2³⁺⁵ = 2⁸
    2. 3² × 3⁴ = 3²⁺⁴ = 3⁶

    Division Law

    When dividing powers with the same base, subtract the exponents.
    aᵐ ÷ aⁿ = aᵐ⁻ⁿ

    Examples

    1. 2⁸ ÷ 2³ = 2⁸⁻³ = 2⁵
    2. 3⁶ ÷ 3² = 3⁶⁻² = 3⁴

    Power Law

    When raising a power to another power, multiply the exponents.
    (aᵐ)ⁿ = aᵐ×ⁿ

    Examples

    1. (2³)⁵ = 2³×⁵ = 2¹⁵
    2. (3²)⁴ = 3²×⁴ = 3⁸

    Zero Index

    Definition

    Any number raised to the power of zero is equal to 1.
    a⁰ = 1

    Examples

    1. 2⁰ = 1
    2. 3⁰ = 1

    Negative Index

    Definition

    A negative index is defined as the reciprocal of the same base raised to the positive power.
    a⁻ᵐ = 1/aᵐ

    Examples

    1. 2⁻¹ = 1/2
    2. 3⁻² = 1/3² = 1/9

    Fractional Index

    Definition

    A fractional index is used to represent roots of numbers.
    aᵐ⁄ⁿ = √(aᵐ)

    Examples

    1. 8¹⁄³ = ∛8
    2. 16¹⁄² = √16 = 4

Square Numbers

A square number is a number that can be expressed as the product of a number and itself. For example:
  • 1 = 1 × 1
  • 4 = 2 × 2
  • 9 = 3 × 3

Questions

  1. What is the value of 2³?
  2. What is the value of 3²?
  3. What is the value of 2⁰?
  4. Simplify the expression: 2³ × 2²
  5. Simplify the expression: 3² ÷ 3¹
  6. What is the square root of 16?
  7. What is the cube root of 27?
  8. Simplify the expression: (2³)²
  9. Simplify the expression: 3² × 3³
  10. What is the value of 2⁵?

Answers

  1. 2³ = 8
  2. 3² = 9
  3. 2⁰ = 1
  4. 2³ × 2² = 2⁵ = 32
  5. 3² ÷ 3¹ = 3¹ = 3
  6. √16 = 4
  7. ∛27 = 3
  8. (2³)² = 2⁶ = 64
  9. 3² × 3³ = 3⁵ = 243
  10. 2⁵ = 32

  1. Simplify the expression: 2² × 2⁴
  2. What is the value of 3⁴?
  3. Simplify the expression: (3²)³
  4. What is the square root of 25?
  5. Simplify the expression: 2³ ÷ 2²

Answers

  1. 2² × 2⁴ = 2⁶ = 64
  2. 3⁴ = 81
  3. (3²)³ = 3⁶ = 729
  4. √25 = 5
  5. 2³ ÷ 2² = 2¹ = 2

Square Numbers Table

Square NumberSquare Root
11
42
93
164
255
366
497
648
819
10010

Cube Numbers Table

Cube NumberCube Root
11
82
273
644
1255
2166
3437
5128
7299
100010


Indices

Positive Indices

Indices are used to show that a number is being multiplied by itself. For example:
  • 3² means 3 × 3
  • 4³ means 4 × 4 × 4

Base, Exponent, and Power

  • The base is the number being multiplied.
  • The exponent or index is the number of times the base is multiplied.
  • The power is the result of multiplying the base by itself as many times as the exponent.

Worked Examples

  1. Write the following in exponential or index notation:
a) 2 × 2 × 2 × 2 × 2 × 2 × 2:
b) 7 × 7 × 7 × 7:
c) 5 × 5 × 5 × 5 × 5:
  1. Write the following in expanded notation:
a) 2⁵:
b) 10⁶:
  1. Write 729 as a power of each number:
a) 27:
b) 9:
c) 3:
  1. Write 216 as a product of the powers of prime numbers.

Questions

  1. What is the value of 3²?
  2. What is the value of 4³?
  3. Write 2 × 2 × 2 × 2 × 2 in exponential notation.
  4. Write 10⁶ in expanded notation.
  5. Write 729 as a power of 3.

Answers

  1. 3² = 9
  2. 4³ = 64
  3. 2 × 2 × 2 × 2 × 2 = 2⁵
  4. 10⁶ = 10 × 10 × 10 × 10 × 10 × 10
  5. 729 = 3⁶

Additional Questions

  1. Write 5 × 5 × 5 × 5 × 5 in exponential notation.
  2. Write 2⁷ in expanded notation.
  3. Write 216 as a product of the powers of prime numbers.
  4. What is the value of 2⁴?
  5. Write 10⁵ in expanded notation.

Answers

  1. 5 × 5 × 5 × 5 × 5 = 5⁵
  2. 2⁷ = 2 × 2 × 2 × 2 × 2 × 2 × 2
  3. 216 = 2³ × 3³
  4. 2⁴ = 16
  5. 10⁵ = 10 × 10 × 10 × 10 × 10

Exponential Notation Table

Exponential NotationExpanded NotationValue
2²2 × 24
3²3 × 39
4²4 × 416
5²5 × 525
2³2 × 2 × 28
3³3 × 3 × 327
4³4 × 4 × 464
5³5 × 5 × 5125


Laws of Indices

Multiplication Law

When multiplying two numbers with the same base, add the exponents.
aᵐ × aⁿ = aᵐ⁺ⁿ

Division Law

When dividing two numbers with the same base, subtract the exponents.
aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Power Law

When raising a power to another power, multiply the exponents.
(aᵐ)ⁿ = aᵐ⁺ⁿ

Worked Examples

  1. Simplify the expression: 2³ × 2²
  2. Simplify the expression: 3² ÷ 3¹
  3. Simplify the expression: (2²)³

Questions

  1. What is the result of 2³ × 2²?
  2. What is the result of 3² ÷ 3¹?
  3. What is the result of (2²)³?

Answers

  1. 2³ × 2² = 2⁵ = 32
  2. 3² ÷ 3¹ = 3¹ = 3
  3. (2²)³ = 2⁶ = 64

Additional Questions

  1. Simplify the expression: 4² × 4³
  2. Simplify the expression: 2⁵ ÷ 2²
  3. Simplify the expression: (3³)²

Answers

  1. 4² × 4³ = 4⁵ = 1024
  2. 2⁵ ÷ 2² = 2³ = 8
  3. (3³)² = 3⁶ = 729

Exponential Notation

Writing Expressions in Exponential Notation

  1. Write the following in exponential notation:
a) 3 × 3 × 3:
b) 12 × 12 × 12 × 12 × 12:
c) 10 × 10 × 10:
  1. Write the following in expanded notation:
a) 7³:
b) 5⁴:
c) 2⁶:

Writing Numbers as Powers of Prime Numbers

  1. Write the following numbers as powers of prime numbers:
a) 8:
b) 27:
c) 49:
d) 121:
e) 125:

Exponential Notation Examples

  1. Write the following in exponential notation:
a) axaxa:
b) m × m × m × m:
c) b × b × b × b × b:
  1. Write 64 in the following ways:
a) power of 8:
b) power of 4:
c) power of a prime number:

Products of Powers of Prime Numbers

  1. Write the following numbers as products of powers of prime numbers:
a) 36:
b) 72:
c) 100:
d) 108:
e) 144:
f) 200:
g) 400:
h) 500:
i) 800:
j) 900:

Zero Indices

Multiplying Powers with the Same Base

  1. Multiply the following powers with the same base:
a) 2² × 2³:
b) 3⁴ × 3²:
c) 4³ × 4⁵:

Division of Powers with the Same Base

  1. Divide the following powers with the same base:
a) 2⁵ ÷ 2²:
b) 3⁶ ÷ 3³:
c) 4⁸ ÷ 4⁴:

Questions

  1. What is the exponential notation for 3 × 3 × 3?
  2. What is the expanded notation for 7³?
  3. Write 8 as a power of a prime number.
  4. Write 64 as a power of 8.
  5. Write 729 as a product of powers of prime numbers.

Answers

  1. 3 × 3 × 3 = 3³
  2. 7³ = 7 × 7 × 7
  3. 8 = 2³
  4. 64 = 8²
  5. 729 = 3⁶

Laws of Indices

Multiplication Law

When multiplying two numbers with the same base, add the exponents.
aᵐ × aⁿ = aᵐ⁺ⁿ

Division Law

When dividing two numbers with the same base, subtract the exponents.
aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Power Law

When raising a power to another power, multiply the exponents.
(aᵐ)ⁿ = aᵐ⁺ⁿ

Worked Examples

  1. Simplify the expression: 2³ × 2²
  2. Simplify the expression: 3² ÷ 3¹
  3. Simplify the expression: (2²)³

Questions

  1. What is the result of 2³ × 2²?
  2. What is the result of 3² ÷ 3¹?
  3. What is the result of (2²)³?

Answers

  1. 2³ × 2² = 2⁵ = 32
  2. 3² ÷ 3¹ = 3¹ = 3
  3. (2²)³ = 2⁶ = 64

Additional Questions

  1. Simplify the expression: 4² × 4³
  2. Simplify the expression: 2⁵ ÷ 2²
  3. Simplify the expression: (3³)²

Answers

  1. 4² × 4³ = 4⁵ = 1024
  2. 2⁵ ÷ 2² = 2³ = 8
  3. (3³)² = 3⁶ = 729

Zero Indices

Definition

Any number raised to the power of zero is equal to 1.
a⁰ = 1

Examples

  1. 2⁰ = 1
  2. 3⁰ = 1
  3. 4⁰ = 1

Questions

  1. What is the value of 2⁰?
  2. What is the value of 3⁰?
  3. What is the value of 4⁰?

Answers

  1. 2⁰ = 1
  2. 3⁰ = 1
  3. 4⁰ = 1

Negative Indices

Definition

A negative index is defined as the reciprocal of the same base raised to the positive power.
a⁻ᵐ = 1/aᵐ

Examples

  1. 2⁻¹ = 1/2
  2. 3⁻² = 1/3² = 1/9
  3. 4⁻³ = 1/4³ = 1/64

Questions

  1. What is the value of 2⁻¹?
  2. What is the value of 3⁻²?
  3. What is the value of 4⁻³?

Answers

  1. 2⁻¹ = 1/2
  2. 3⁻² = 1/9
  3. 4⁻³ = 1/64

Fractional Indices

Definition

A fractional index is defined as the nth root of the same base raised to the power of m.
aᵐ⁄ⁿ = √(aᵐ)

Examples

  1. 2¹⁄² = √2
  2. 3²⁄³ = (√(3²)) = √9 = 3
  3. 4³⁄⁴ = (√(4³)) = √64 = 4

Questions

  1. What is the value of 2¹⁄²?
  2. What is the value of 3²⁄³?
  3. What is the value of 4³⁄⁴?

Answers

  1. 2¹⁄² = √2
  2. 3²⁄³ = 3
  3. 4³⁄⁴ = 4

Fractional Indices

Fractional indices are used to represent roots of numbers. For example:
  • 8¹⁄³ represents the cube root of 8
  • 16¹⁄² represents the square root of 16

Rules for Fractional Indices

  1. Square Root: a¹⁄² = √a
  2. Cube Root: a¹⁄³ = ∛a
  3. Fourth Root: a¹⁄⁴ = √√a
  4. General Rule: aᵐ⁄ⁿ = √(aᵐ)

Examples

  1. Simplify: 9¹⁄²
9¹⁄² = √9 = 3
  1. Simplify: 27¹⁄³
27¹⁄³ = ∛27 = 3
  1. Simplify: 16¹⁄⁴
16¹⁄⁴ = √√16 = √4 = 2

Questions

  1. Simplify: 25¹⁄²
  2. Simplify: 64¹⁄³
  3. Simplify: 81¹⁄⁴

Answers

  1. 25¹⁄² = √25 = 5
  2. 64¹⁄³ = ∛64 = 4
  3. 81¹⁄⁴ = √√81 = √9 = 3

Standard Form (Scientific Notation)

Definition

Standard form, also known as scientific notation, is a way of expressing very large or very small numbers in a more compact form.

Format

The standard form is written as:
a × 10ⁿ
where:
  • a is a number between 1 and 10
  • 10ⁿ is the power of 10

Examples

  1. 456,000 = 4.56 × 10⁵
  2. 0.000456 = 4.56 × 10⁻⁴

Questions

  1. Write 34,500 in standard form.
  2. Write 0.0000345 in standard form.

Answers

  1. 34,500 = 3.45 × 10⁴
  2. 0.0000345 = 3.45 × 10⁻⁵

Converting Between Standard Form and Ordinary Numbers

Examples

  1. Convert 4.23 × 10³ to an ordinary number:
4.23 × 10³ = 4,230
  1. Convert 2.56 × 10⁻⁴ to an ordinary number:
2.56 × 10⁻⁴ = 0.000256

Questions

  1. Convert 3.45 × 10⁴ to an ordinary number.
  2. Convert 1.23 × 10⁻⁵ to an ordinary number.

Answers

  1. 3.45 × 10⁴ = 34,500
  2. 1.23 × 10⁻⁵ = 0.0000123

Operations with Numbers in Standard Form

Multiplication

When multiplying numbers in standard form, multiply the coefficients and add the exponents.
(a × 10ⁿ) × (b × 10ᵐ) = (a × b) × 10ⁿ⁺ᵐ

Division

When dividing numbers in standard form, divide the coefficients and subtract the exponents.
(a × 10ⁿ) ÷ (b × 10ᵐ) = (a ÷ b) × 10ⁿ⁻ᵐ

Examples

  1. Multiply 2.3 × 10³ and 4.5 × 10²:
(2.3 × 10³) × (4.5 × 10²) = (2.3 × 4.5) × 10³⁺² = 10.35 × 10⁵
  1. Divide 6.7 × 10⁴ by 2.1 × 10²:
(6.7 × 10⁴) ÷ (2.1 × 10²) = (6.7 ÷ 2.1) × 10⁴⁻² = 3.19 × 10²

Questions

  1. Multiply 3.2 × 10³ and 2.1 × 10².
  2. Divide 8.5 × 10⁵ by 3.4 × 10³.

Answers

  1. (3.2 × 10³) × (2.1 × 10²) = (3.2 × 2.1) × 10³⁺² = 6.72 × 10⁵
  2. (8.5 × 10⁵) ÷ (3.4 × 10³) = (8.5 ÷ 3.4) × 10⁵⁻³ = 2.5 × 10²

Laws of Indices

Multiplication Law

When multiplying powers with the same base, add the exponents.
aᵐ × aⁿ = aᵐ⁺ⁿ

Examples

  1. 2³ × 2⁵ = 2³⁺⁵ = 2⁸
  2. 3² × 3⁴ = 3²⁺⁴ = 3⁶

Division Law

When dividing powers with the same base, subtract the exponents.
aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Examples

  1. 2⁸ ÷ 2³ = 2⁸⁻³ = 2⁵
  2. 3⁶ ÷ 3² = 3⁶⁻² = 3⁴

Power Law

When raising a power to another power, multiply the exponents.
(aᵐ)ⁿ = aᵐ⁺ⁿ

Examples

  1. (2³)⁵ = 2³⁺⁵ = 2¹⁵
  2. (3²)⁴ = 3²⁺⁴ = 3⁸

Zero Index

Definition

Any number raised to the power of zero is equal to 1.
a⁰ = 1

Examples

  1. 2⁰ = 1
  2. 3⁰ = 1

Negative Index

Definition

A negative index is defined as the reciprocal of the same base raised to the positive power.
a⁻ᵐ = 1/aᵐ

Examples

  1. 2⁻¹ = 1/2
  2. 3⁻² = 1/3² = 1/9

Fractional Index

Definition

A fractional index is used to represent roots of numbers.
aᵐ⁄ⁿ = √(aᵐ)

Examples

  1. 8¹⁄³ = ∛8
  2. 16¹⁄² = √16 = 4

Dividing Powers with the Same Base

When dividing powers with the same base, subtract the exponents.
aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Examples

  1. 2⁵ ÷ 2² = 2⁵⁻² = 2³
  2. 3⁶ ÷ 3⁴ = 3⁶⁻⁴ = 3²

Worked Examples

  1. Simplify: 2⁵ ÷ 2²
  2. Simplify: 3⁶ ÷ 3⁴

Answers

  1. 2⁵ ÷ 2² = 2³ = 8
  2. 3⁶ ÷ 3⁴ = 3² = 9

Zero Index

Definition

Any number raised to the power of zero is equal to 1.
a⁰ = 1

Examples

  1. 2⁰ = 1
  2. 3⁰ = 1

Negative Index

Definition

A negative index is defined as the reciprocal of the same base raised to the positive power.
a⁻ᵐ = 1/aᵐ

Examples

  1. 2⁻¹ = 1/2
  2. 3⁻² = 1/3² = 1/9

Fractional Index

Definition

A fractional index is used to represent roots of numbers.
aᵐ⁄ⁿ = √(aᵐ)

Examples

  1. 8¹⁄³ = ∛8
  2. 16¹⁄² = √16 = 4

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