Unit 0 - Required Prior Knowledge
Required Prior Knowledge
Questions
Without using your calculator, evaluate the following:
a) \(\left(-1\right)^{5} \)
b) \(\left(-1\right)^{6}\)
c) \(\left(-1\right)^{14}\)
d) \(\left(-1\right)^{19}\)
e) \(\left(-1\right)^{8}\)
f) \(-1^{8}\)
g) \(-\left(-1\right)^{8}\)
h) \(\left(-2\right)^{5}\)
i) \(-2^{5}\)
j) \(-\left(-2\right)^{6}\)
k) \(\left(-5\right)^{4}\)
l) \(-\left(-5\right)^{4}\)
Solutions
Get Ready
Questions
Find the values of \(3^{1},3^{2},3^{3},3^{4},3^{5},…\)
Look for patterns.
Hence find the last digit of \(3^{101}\).
Solutions
Notes
The laws of indices (or exponents or powers) are a set of rules that tell us how to work with indices in order to simplify expressions. These are considered prior knowledge, but a summary is given below. This includes examples and non-examples (common mistakes) for each rule.
Examples and Your Turns
Your Turn
Evaluate each of these using the laws of indices:
a) $$\left(\frac{27}{8}\right)^{\frac{2}{3}}$$
b) $$144^{-\frac{1}{2}}$$
c) $$-125^{\frac{1}{3}}$$
Your Turn
Simplify the expression$$\frac{20\times 9^{2n+1}}{5\times 3^{4n+1}}$$
Your Turn
Simplify the expression$$\left(\frac{a^{\frac{2}{3}}\sqrt{b^{-1}}}{b\sqrt[3]{a^{-2}}}\div \sqrt{\frac{a\sqrt{b^{-4}}}{b\sqrt{a^{-2}}}}\right)^{6}$$
Your Turn
Simplify the expression$$\left(\frac{a^{p-q}}{\sqrt[q]{a^{q^{2}-pq}}}\times a^{2\left(p-q\right)}\right)^{n}$$
Your Turn
Simplify the expression$$\left(x^{\frac{a}{b}y^{-1}}\right)^{b}\div \left(\frac{x^{a^{2}-b^{2}}}{y^{ab+b^{2}}}\right)^{\frac{1}{a+b}}$$
Your Turn
Simplify the expression$$\left(\frac{x^{-2}y^{3}}{x^{3}y^{-2}}\right)^{\frac{1}{5}}\times \left(\frac{y^{3}x^{-3}}{x^{3}y^{-3}}\right)^{-1}$$
Your Turn
Simplify the expression$$\left(\frac{y^{-3}}{x^{\frac{2}{7}}z^{-1}}\right)^{-\frac{3}{2}}\times \left(\frac{y^{\frac{14}{3}}x^{-1}}{z^{-\frac{21}{4}}}\right)^{\frac{2}{7}}$$
Your Turn
Simplify the expression$$\frac{2^{n}\times \left(2^{n-1}\right)^{n}}{2^{n+1}\times 2^{n-1}}\times \frac{1}{4^{-n}}$$
Your Turn
Simplify the expression$$\frac{2^{n+1}}{\left(2^{n}\right)^{n-1}}\div \frac{4^{n+1}}{\left(2^{n-1}\right)^{n+1}}$$
Example
Factorise$$2^{n+3}+2^{n}$$
Your Turn
Factorise $$9^{x}+4\left(3^{x}\right)+4$$
Example
Solve$$3^{x}=3^{1-x}=4$$
Your Turn
Solve $$4^{x}+2^{x}-20=0$$
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.