What are Powers in Math? [pptime time="0:01"](0:01) [/pptime]

A power is a way of writing repeated multiplication of the same number. They help express large and small numbers efficiently. In general, a power is written as: $$ a^n $$ where: $ a $ is the base . The base is the number being multiplied. $ n $ is the index ( exponent ). The index (exponent) tells us how many times the base is multiplied by itself. For example: $$ 3^4 = 3 \times 3 \times 3 \times 3 $$ Here, 3 is the base. 4 is the index, meaning 3 is multiplied by itself 4 times. This is pronounced as "3 to the power of 4".

What are Key Properties of Powers? [pptime time="1:00"](1:00) [/pptime]

Understanding the following properties will help you develop a strong grasp of exponents and indices. The Base and Exponent Cannot Be Swapped. $$ \text{For example, } 3^4 \neq 4^3 $$ Any Number to the Power of 1 Equals Itself. $$ a^1 = a $$ Any Number (Except 0) to the Power of 0 Equals 1 $$ a^0 = 1 \quad \text{where } a \neq 0 $$

What are Square Numbers and Cube Numbers? [pptime time="1:30"](1:30) [/pptime]

Two of the most common powers are squares (index 2) and cubes (index 3). Square Numbers: A square number is a number that results from multiplying a whole number by itself. $$ n^2 = n \times n $$ Common Square Numbers: $$ 1^2 = 1, \quad 2^2 = 4, \quad 3^2 = 9, \quad 4^2 = 16, \quad 5^2 = 25, \quad 6^2 = 36 $$ Cube Numbers: A cube number is a number that results from multiplying a whole number by itself three times. $$ n^3 = n \times n \times n $$ Common Cube Numbers: $$ 1^3 = 1, \quad 2^3 = 8, \quad 3^3 = 27, \quad 4^3 = 64, \quad 5^3 = 125 $$ Recognizing square and cube numbers helps you simplify expressions and solve equations more easily.

How to Apply BIDMAS with Powers (Indices)? [pptime time="2:30"](2:30) [/pptime]

In BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction), indices (exponents) are calculated after brackets but before all basic operations. Example 1: Brackets Before Indices When an expression has brackets, evaluate the brackets first: $$ (2 \times 5)^2 $$ Step 1: Solve inside the brackets: $$ (10)^2 $$ Step 2: Apply the exponent: $$ 100 $$ Example 2: Indices Before Multiplication If there are no brackets around the multiplication, apply the exponent first: $$ 2 \times 5^2 $$ Step 1: Evaluate the exponent first: $$ 2 \times 25 $$ Step 2: Perform the multiplication: $$ 50 $$

Powers and Indices - Base, Exponent, Square & Cube Numbers

Q: How to Use Exponents to Simplify Expressions? [pptime time="2:05"](2:05) [/pptime]

When multiplying the same number multiple times , you can use exponents (indices) to write the expression in a simplified form. For example: $$ 4 \times 3 \times 3 \times 4 \times 4 $$ We can group the same factors using the commutative and associative laws: $$ (4 \times 4 \times 4) \times (3 \times 3) $$ Since each base is repeated, we can write it using exponents: $$ 4^3 \times 3^2 $$ By using exponents for repeated multiplication, you make expressions easier to read, write, and work with.

What are Powers in Math? (0:01)

A power is a way of writing repeated multiplication of the same number. They help express large and small numbers efficiently. In general, a power is written as:

$$ a^n $$

where:

$ a $ is the base. The base is the number being multiplied.
$ n $ is the index (exponent). The index (exponent) tells us how many times the base is multiplied by itself.

For example:
$$ 3^4 = 3 \times 3 \times 3 \times 3 $$

Here, 3 is the base.
4 is the index, meaning 3 is multiplied by itself 4 times.
This is pronounced as “3 to the power of 4”.

What are Key Properties of Powers? (1:00)

Understanding the following properties will help you develop a strong grasp of exponents and indices.

The Base and Exponent Cannot Be Swapped.

$$ \text{For example, } 3^4 \neq 4^3 $$

Any Number to the Power of 1 Equals Itself.

$$ a^1 = a $$

Any Number (Except 0) to the Power of 0 Equals 1

$$ a^0 = 1 \quad \text{where } a \neq 0 $$

What are Square Numbers and Cube Numbers? (1:30)

Two of the most common powers are squares (index 2) and cubes (index 3).

Square Numbers:

A square number is a number that results from multiplying a whole number by itself.

$$ n^2 = n \times n $$

Common Square Numbers:

$$ 1^2 = 1, \quad 2^2 = 4, \quad 3^2 = 9, \quad 4^2 = 16, \quad 5^2 = 25, \quad 6^2 = 36 $$

Cube Numbers:

A cube number is a number that results from multiplying a whole number by itself three times.

$$ n^3 = n \times n \times n $$

Common Cube Numbers:

$$ 1^3 = 1, \quad 2^3 = 8, \quad 3^3 = 27, \quad 4^3 = 64, \quad 5^3 = 125 $$

Recognizing square and cube numbers helps you simplify expressions and solve equations more easily.

How to Use Exponents to Simplify Expressions? (2:05)

When multiplying the same number multiple times, you can use exponents (indices) to write the expression in a simplified form.

For example:

$$ 4 \times 3 \times 3 \times 4 \times 4 $$

We can group the same factors using the commutative and associative laws:

$$ (4 \times 4 \times 4) \times (3 \times 3) $$

Since each base is repeated, we can write it using exponents:

$$ 4^3 \times 3^2 $$

By using exponents for repeated multiplication, you make expressions easier to read, write, and work with.

How to Apply BIDMAS with Powers (Indices)? (2:30)

In BIDMAS (Brackets, Indices, Division, Multiplication, Addition, Subtraction),
indices (exponents) are calculated after brackets but before all basic operations.

Example 1: Brackets Before Indices
When an expression has brackets, evaluate the brackets first:

$$ (2 \times 5)^2 $$