# Grade 11

2022 TERM 1

Week 1Week 2Week 3Week 4Week 5Week 6Week 7Week 8Week 9Week 10
Exponents and SurdsEquations and InequalitiesEuclidean GeometryTrigonometry

## Algebra – Exponential Equations and Expressions

• The number system:
• $$\mathbb{N}$$: natural numbers are $$\{1; \; 2; \; 3; \; \ldots\}$$
• $$\mathbb{N}_0$$: whole numbers are $$\{0; \; 1; \; 2; \; 3; \; \ldots\}$$
• $$\mathbb{Z}$$: integers are $$\{\ldots; \; -3; \; -2; \; -1; \; 0; \; 1; \; 2; \; 3; \; \ldots\}$$
• $$\mathbb{Q}$$: rational numbers are numbers which can be written as $$\frac{a}{b}$$ where $$a$$ and
$$b$$ are integers and $$b\ne 0$$, or as a terminating or recurring decimal number.
• $$\mathbb{Q}’$$: irrational numbers are numbers that cannot be written as a fraction with the numerator
and denominator as integers. Irrational numbers also include decimal numbers that neither terminate nor
recur.
• $$\mathbb{R}$$: real numbers include all rational and irrational numbers.
• $$\mathbb{R}’$$: non-real numbers or imaginary numbers are numbers that are not real.
• Definitions:
• $${a}^{n}=a\times a\times a\times \cdots \times a \left(n \text{ times}\right) \left(a\in \mathbb{R},n\in \mathbb{N}\right)$$
• $${a}^{0}=1$$ ($$a \ne 0$$ because $$0^0$$ is undefined)
• $${a}^{-n}=\frac{1}{{a}^{n}}$$ ($$a \ne 0$$ because $$\dfrac{1}{0}$$ is undefined)
• Laws of exponents:
• $${a}^{m} \times {a}^{n}={a}^{m+n}$$
• $$\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}$$
• $${\left(ab\right)}^{n}={a}^{n}{b}^{n}$$
• $${\left(\frac{a}{b}\right)}^{n}=\dfrac{{a}^{n}}{{b}^{n}}$$
• $${\left({a}^{m}\right)}^{n}={a}^{mn}$$

where $$a > 0$$, $$b > 0$$ and $$m, n \in \mathbb{Z}$$.

• Rational exponents and surds:
• If $$r^n = a$$, then $$r = \sqrt[n]{a} \quad (n \geq 2)$$
• $$a^{\frac{1}{n}} = \sqrt[n]{a}$$
• $$a^{-\frac{1}{n}} = (a^{-1})^{\frac{1}{n}} = \sqrt[n]{\dfrac{1}{a}}$$
• $$a^{\frac{m}{n}} = (a^{m})^{\frac{1}{n}} = \sqrt[n]{a^m}$$

where $$a > 0$$, $$r > 0$$ and $$m,n \in \mathbb{Z}$$, $$n \ne 0$$.

• Simplification of surds:
• $$\sqrt[n]{a}\sqrt[n]{b} = \sqrt[n]{ab}$$
• $$\sqrt[n]{\dfrac{a}{b}} = \dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$$
• $$\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}$$

## The Quadratic formula

It is not always possible to solve a quadratic equation by factorisation. The quadratic formula provides an easy and fast way to solve quadratic equations.

$x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$

For any quadratic equation $$ax^2+bx+c = 0$$ we can determine two roots
$x = \dfrac{-b + \sqrt{b^2-4ac}}{2a} \text{ and } x = \dfrac{-b – \sqrt{b^2-4ac}}{2a}$

It is important to notice that the expression $${b}^{2}-4ac$$ must be greater than or equal to zero for the roots
of the quadratic to be real. If the expression under the square root sign is less than zero, then the roots are
non-real (imaginary).