Core Chapter 3: Surds and Exponents

Question 1

Distinguish between a radical and a surd

Answer 1

Radical: any non-negative number written under the radical sign (√), where
√a x √a = a
a ≥ 0
√a ≥ 0

Surd: real, irrational radical

Question 2

State the 2 rules of radicals

Answer 2

1) √a x b = √a x √b
2) √a / b = √a / √b

Question 3

Express radicals in their simplest form

Eg. √72

Answer 3

Simplest form: when the number under the radical is the smallest possible integer (make use of rules of radicals to simplify)

Eg. √72 = √2x36 = √2 x √36 = 6√2

Question 4

Add and subtract radicals with different integers under the radical sign
(Eg. 2√75 - 5√27)

Answer 4

Express both radicals such that the same integer is under the radical sign (usually in simplest form)

Eg. 2√75 - 5√27 = 2x√25x√3 - 5x√9x√3
= 10√3 - 15√3
= -5√3

Question 5

Rationalise fractions to be without the surd in the denominator
(b/√a or c/a+√b)

Answer 5

b/√a:
- Multiply fraction by √a/√a
(√a/√a = 1 and does not change the value of the fraction, but allows the denominator to become a non-surd, as √ax√a=a)
Eg. 6/√5 x √5/√5 = 6√5/5

c/a+√b:
- Multiply fraction by its conjugate
(a-√b/a-√b), rationalising the surd denominator as
(a+√b)x(a-√b) = a²-b²
and
(√b)² = b
Eg. 5/3-√2 x 3+√2/3+√2 = 15+5√2/9-2
= 15+5√2/7

Question 6

Describe what exponents are

Answer 6

aⁿ is the product of n factors of a (n>0)
- a: base
- n: power/exponent/index
(when a<0 and is raised to an odd power, it remains negative. but if a is raised to a positive power, it becomes positive)
(when a>0, it remains positive regardless of whether it is raised to a positive or negative power)

Question 7

State the 7 laws of exponents

Answer 7

1) aᵐxaⁿ = aᵐ⁺ⁿ
2) aᵐ/aⁿ = aᵐ⁻ⁿ
3) (aᵐ)ⁿ = aᵐⁿ
4) (ab)ᵐ = aᵐbᵐ
5) (a/b)ᵐ = aᵐ/bᵐ
6) a⁰=1
7) a⁻ⁿ = 1/aⁿ