Grade 12Study
Mathematics Algebra
Notes
Topic Page
1. Exponents & Surds 2
2. Equations 3
3. Inequalities 4
4. Number Patterns & Sequences 5
5. Functions & Graphs 6
6. Financial Mathematics 7
, 1. Exponents & Surds
Rules for working with powers and square roots
Laws of Exponents
Multiply same base Divide same base
a^m x a^n = a^(m+n) a^m / a^n = a^(m-n)
Power of a power Negative exponent
(a^m)^n = a^(mn) a^(-n) = 1/a^n
Zero exponent Fraction exponent
a^0 = 1 a^(1/n) = nth-root(a)
EXAMPLE — Simplify using exponent laws
2^3 x 2^5 = 2^(3+5) = 2^8 = 256
x^(-2) = 1/x^2 (flip it to make the exponent positive)
Surds (Irrational Roots)
A surd is a root that does not simplify to a whole number, such as sqrt(2) or sqrt(5).
Multiply surds Divide surds
sqrt(a) x sqrt(b) = sqrt(ab) sqrt(a) / sqrt(b) = sqrt(a/b)
Simplify Add (like terms)
sqrt(12) = sqrt(4x3) = 2*sqrt(3) 3*sqrt(2) + 5*sqrt(2) = 8*sqrt(2)
TIP: To rationalise the denominator (remove the surd from below), multiply top and bottom by
the surd.
Example: 1/sqrt(3) x sqrt(3)/sqrt(3) = sqrt(3)/3
Mathematics Algebra
Notes
Topic Page
1. Exponents & Surds 2
2. Equations 3
3. Inequalities 4
4. Number Patterns & Sequences 5
5. Functions & Graphs 6
6. Financial Mathematics 7
, 1. Exponents & Surds
Rules for working with powers and square roots
Laws of Exponents
Multiply same base Divide same base
a^m x a^n = a^(m+n) a^m / a^n = a^(m-n)
Power of a power Negative exponent
(a^m)^n = a^(mn) a^(-n) = 1/a^n
Zero exponent Fraction exponent
a^0 = 1 a^(1/n) = nth-root(a)
EXAMPLE — Simplify using exponent laws
2^3 x 2^5 = 2^(3+5) = 2^8 = 256
x^(-2) = 1/x^2 (flip it to make the exponent positive)
Surds (Irrational Roots)
A surd is a root that does not simplify to a whole number, such as sqrt(2) or sqrt(5).
Multiply surds Divide surds
sqrt(a) x sqrt(b) = sqrt(ab) sqrt(a) / sqrt(b) = sqrt(a/b)
Simplify Add (like terms)
sqrt(12) = sqrt(4x3) = 2*sqrt(3) 3*sqrt(2) + 5*sqrt(2) = 8*sqrt(2)
TIP: To rationalise the denominator (remove the surd from below), multiply top and bottom by
the surd.
Example: 1/sqrt(3) x sqrt(3)/sqrt(3) = sqrt(3)/3
