Unit 0 - Required Prior Knowledge
Required Prior Knowledge
Questions
Without using a calculator, work out
a) \(16^{\frac{1}{2}} \)
b) \(100^{\frac{1}{2}}\)
c) \(27^{\frac{1}{3}}\)
d) \(64^{\frac{1}{3}}\)
e) \(8^{\frac{2}{3}}\)
f) \(49^{\frac{3}{2}}\)
Solutions
Get Ready
Questions
The two basic laws of surds are$$\sqrt{a}\times\sqrt{b}=\sqrt{ab}$$and$$\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$$Work out
a) \(\sqrt{3}\times\sqrt{5}=\)
b) \(2\times\sqrt{3}=\)
c) \(\sqrt{5}\times 3=\)
d) \(\sqrt{2}\times\sqrt{8}=\)
e) \(\sqrt{3}\times\sqrt{3}=\)
f) \(2\sqrt{3}\times 2\sqrt{5}=\)
g) \(3\sqrt{2}\times 3\sqrt{2}=\)
h) \(\sqrt{18}\times 4\sqrt{2}=\)
i) \(\sqrt{12}\div\sqrt{3}=\)
k) \(\sqrt{44}\div\sqrt{4}=\)
l) \(\sqrt{\frac{16}{81}}=\)
m) \(\sqrt{\frac{2}{9}}=\)
n) \(6\sqrt{5}\div 3\sqrt{5}=\)
o) \(4\sqrt{15}\div 2\sqrt{5}=\)
p) \(3\sqrt{\frac{16}{9}}=\)
q) \(\sqrt{\frac{18}{27}}=\)
Solutions
Notes
Surds are numbers in roots which cannot be simplified down to a rational number.
To simplify a surd we look for the largest square number that is a factor of the number inside the root, and create a poduct of two roots. We then simplify the root with the square number.
Examples and Your Turns
Example
Simplify$$\sqrt{27}$$
Your Turn
Simplify$$\sqrt{98}$$
-
$$\begin{align}\sqrt{98}&=\sqrt{49}\sqrt{2}\\&=7\sqrt{2}\end{align}$$
Your Turn
Simplify$$\sqrt{72}$$
-
$$\begin{align}\sqrt{72}&=\sqrt{36}\sqrt{2}\\&=6\sqrt{2}\end{align}$$
Your Turn
Simplify$$\sqrt{32}$$
-
$$\begin{align}\sqrt{32}&=\sqrt{16}\sqrt{2}\\&=4\sqrt{2}\end{align}$$
Your Turn
Simplify$$2\sqrt{50}$$
Your Turn
Simplify$$4\sqrt{12}$$
-
$$\begin{align}4\sqrt{12}&=4\sqrt{4}\sqrt{3}\\&=4\times 2\sqrt{3}\\&=8\sqrt{3}\end{align}$$
Notes
To add or subtract surds, we simplify all surds and then ‘collect’ like surds, in a similar way to working with algebra.
Examples and Your Turns
Example
Simplify$$2\sqrt{5}+\sqrt{5}$$
Your Turn
Simplify$$7\sqrt{7}+7\sqrt{7}$$
-
$$7\sqrt{7}+7\sqrt{7}=14\sqrt{7}$$
Example
Simplify$$\sqrt{2}+\sqrt{8}$$
Your Turn
Simplify$$\sqrt{3}+\sqrt{48}$$
-
$$\begin{align}\sqrt{3}+\sqrt{48}&=\sqrt{3}+\sqrt{16}\sqrt{3}\\&=\sqrt{3}+4\sqrt{3}\\&=5\sqrt{3}\end{align}$$
Your Turn
Simplify$$\sqrt{300}+5\sqrt{75}$$
Your Turn
Simplify$$3\sqrt{12}+\sqrt{27}$$
-
$$\begin{align}3\sqrt{12}+\sqrt{27}&=3\sqrt{4}\sqrt{3}+\sqrt{9}\sqrt{3}\\&=6\sqrt{3}+3\sqrt{3}\\&=9\sqrt{3}\end{align}$$
Notes
We can expand expressions with surds and brackets just like with algebra.
Examples and Your Turns
Example
Simplify$$\sqrt{2}\left(3+\sqrt{2}\right)$$
Your Turn
Simplify$$\sqrt{5}\left(2+\sqrt{5}\right)$$
-
$$\sqrt{5}\left(2+\sqrt{5}\right)=5\sqrt{2}+5$$
Your Turn
Simplify$$3\sqrt{2}\left(2+2\sqrt{2}\right)$$
Your Turn
Simplify$$4\sqrt{5}\left(3+2\sqrt{5}\right)$$
-
$$\begin{align}4\sqrt{5}\left(3+2\sqrt{5}\right)&=12\sqrt{5}+4\sqrt{25}\\&=12\sqrt{5}+20\end{align}$$
Example
Simplify$$\left(\sqrt{8}+3\right)\left(\sqrt{2}+5\right)$$
Your Turn
Simplify$$\left(\sqrt{12}+2\right)\left(\sqrt{3}+7\right)$$
-
$$\begin{align}\left(\sqrt{12}+2\right)\left(\sqrt{3}+7\right)&=\sqrt{12}\sqrt{3}+7\sqrt{12}+2\sqrt{3}+14\\&=\sqrt{36}+7\sqrt{4}\sqrt{3}+2\sqrt{3}+14\\&=6+14\sqrt{3}+2\sqrt{3}+14\\&=20+16\sqrt{3}\end{align}$$
Notes
Most of the time we prefer to not have surds in the denominator of a fraction.
We do this by rationalising the denominator.
In essence, we mutliply the fraction by \(1\), and thus do not change the value of the fraction. But we choose how we write the \(1\) in a way that we can then simplify the denominator to a rational number.
Examples and Your Turns
Example
Simplify$$\frac{1}{\sqrt{5}}$$
Your Turn
Simplify$$\frac{1}{\sqrt{3}}$$
-
$$\begin{align}\frac{1}{\sqrt{3}}&=\frac{1}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}\\&=\frac{\sqrt{3}}{\sqrt{3}\sqrt{3}}\\&=\frac{\sqrt{3}}{3}\end{align}$$
Example
Simplify$$\frac{3}{\sqrt{2}}$$
Your Turn
Simplify$$\frac{12}{\sqrt{5}}$$
-
$$\begin{align}\frac{12}{\sqrt{5}}&=\frac{12}{\sqrt{5}}\times\frac{\sqrt{5}}{\sqrt{5}}\\&=\frac{12\sqrt{5}}{\sqrt{5}\sqrt{5}}\\&=\frac{12\sqrt{5}}{5}\end{align}$$
Example
Simplify$$\frac{8}{3\sqrt{2}}$$
Your Turn
Simplify$$\frac{10}{5\sqrt{3}}$$
-
$$\begin{align}\frac{10}{5\sqrt{3}}&=\frac{10}{5\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}\\&=\frac{10\sqrt{3}}{5\sqrt{3}\sqrt{3}}\\&=\frac{10\sqrt{3}}{15}\\&=\frac{2\sqrt{3}}{3}\end{align}$$
Notes
Recall the difference of two squares rule:$$\left(a-b\right)\left(a+b\right)=$$
-
$$\left(a-b\right)\left(a+b\right)=a^{2}-b^{2}$$
If the denominator of a fraction is a sum (or difference) of surds, then we multiply by \(1\) in a form where we can make use of the difference of two squares in the denominator.
Examples and Your Turns
Example
Simplify$$\frac{3}{\sqrt{6}-2}$$
Your Turn
Simplify$$\frac{4}{\sqrt{3}+1}$$
Your Turn
Simplify$$\frac{\sqrt{2}+1}{\sqrt{2}+2}$$
Your Turn
Simplify$$\frac{3+\sqrt{3}}{4-\sqrt{3}}$$
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.