Integers, Roots, Exponents & BEDMAS

Integers

Whole numbers that can be positive, negative, or zero

A number line helps us understand these:

• Moving right → adding

• Moving left → subtracting

Eg.

• Positive integers: 1, 2, 3, …

• Negative integers: –1, –2, –3, …

• Zero: 0

Addition with integers

Same signs → add and keep the sign

Eg. (–3) + (–5) = –8

Different signs → subtract and keep the sign of the number with the larger absolute value

Eg. 7 + (–4) = 3

Subtraction with integers

Change subtraction into addition of the opposite

Eg.

- 5 – (–3) =

5 + 3 =

8

Steps for exam questions:

1. Rewrite subtraction as addition

2. Apply integer addition rules

3. Check if your answer makes sense on a number line

Common mistakes during operations with integers to avoid

• Forgetting that subtracting a negative makes the value larger

• Ignoring the sign of the number with the bigger magnitude

A root

The inverse of a power

• Square root (√): asks “What number multiplied by itself gives this value?”

  • Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, …

• Cube root (∛): asks “What number multiplied three times gives this value?”

  • Perfect cubes: 1, 8, 27, 64, 125, …

Eg.

√25 = 5 because 5 × 5 = 25

∛8 = 2 because 2 × 2 × 2 = 8

Square Roots

When we multiply a number to itself, we square it.

When we figure out what number is already multiplied to itself, we square root it

Cube Roots

When we multiply a number to itself three times, we cube it.

When we figure out what number is already multiplied to itself, we cube root it

Estimating roots

If the number is not a perfect square:

Eg:

• √20

• 4² = 16

• 5² = 25

• √20 lies between 4 and 5

Steps for estimation questions:

1. Find the nearest perfect squares

2. Decide which two integers the root lies between

Exponent

Shows repeated multiplication.

Eg:

• 2³ = 2 × 2 × 2 = 8

Law 1 of exponents (Multiplication)

Add the exponents:

xᵃ × xᵇ = xᵃ⁺ᵇ

Eg:

5² × 5³ = 5⁵

Law 2 of exponents (Division)

Subtract the exponents:

xᵃ ÷ xᵇ = xᵃ⁻ᵇ

Eg:

8⁵ ÷ 8² = 8³

Law 3 of exponents (Power of a power)

Multiply the exponents:

(xᵃ)ᵇ = xᵃᵇ

Eg:

(2³)² = 2⁶

Law 4 of exponents (Negative Exponent)

Divide the number and exponent by one:

a⁻ⁿ = 1 / aⁿ

Eg:

9⁻² = 1 / 9²

Law 5 of exponents (Zero Exponent)

Any non-zero number to the power of zero equals 1:

x⁰ = 1

Eg:

7⁰ = 1

BEDMAS

(Brackets)

Exponents²

Division Multiplication

Addition Subtraction

Nb. Go left to right

8 + 2 × 5 = ?

Multiplication first: 2 × 5 = 10

Addition: 8 + 10 = 18

10 – 2³ + 4 = ?

Exponent: 2³ = 8

Subtraction and addition (left to right): 10 – 8 + 4 = 6

12 – √16 × 2² = ?

Roots and exponents: √16 = 4, 2² = 4

Multiplication: 4 × 4 = 16

Subtraction: 12 – 16 = –4