Exponents

Introduction

In Grade 10 we studied exponential numbers and learnt that there were six laws that made working with exponential numbers easier. There is one law that we did not study in Grade 10. This will be described here.

Laws of exponents

In Grade 10, we worked only with indices that were integers. What happens when the index is not an integer, but is a rational number? This leads us to the final law of exponents,

Exponential law 7: $a^{\frac{m}{n}}=\sqrt[n]{a^m}$

We say that $x$ is an $n$ th root of $b$ if $x^n=b$ and we write $x=\sqrt[n]{b}$ . $n^{\mathrm{th}}$ roots written with the radical symbol, $\sqrt{}$ , are referred to as surds. For example, ${(-1)}^4=1$ , so $-1$ is a 4th root of 1. Using law 6, we notice that

therefore $a^{\frac{m}{n}}$ must be an $n$ th root of $a^m$ . We can therefore say

For example,

A number may not always have a real $n$ th root. For example, if $n=2$ and $a=-1$ , then there is no real number such that $x^2=-1$ because $x^2\ge 0$ for all real numbers $x$ .

It is also possible for more than one $n\mathrm{th}$ root of a number to exist. For example, ${(-2)}^2=4$ and $2^2=4$ , so both $-2$ and 2 are 2nd (square) roots of 4. Usually, if there is more than one root, we choose the positive real solution and move on.

The following videos work through two examples of simplifying expressions.

Applying laws

Use all the laws to:

1. Simplify:
   1. $\left(x^0\right)+5x^0-{(0,25)}^{-0,5}+8^{\frac{2}{3}}$
   2. $s^{\frac{1}{2}}÷s^{\frac{1}{3}}$
   3. $\frac{12m^{\frac{7}{9}}}{8m^{-\frac{11}{9}}}$
   4. ${\left(64m^6\right)}^{\frac{2}{3}}$
2. Re-write the following expression as a power of $x$ :
   $$x\sqrt{x\sqrt{x\sqrt{x\sqrt{x}}}}$$

Exponentials in the real-world

In Grade 10 Finance, you used exponentials to calculate different types of interest, for example on a savings account or on a loan and compound growth.

End of chapter exercises

1. Simplify as far as possible:
   1. $8^{-\frac{2}{3}}$
   2. $\sqrt{16}+8^{-\frac{2}{3}}$
2. Simplify:
   1. ${\left(x^3\right)}^{\frac{4}{3}}$
   2. ${\left(s^2\right)}^{\frac{1}{2}}$
   3. ${\left(m^5\right)}^{\frac{5}{3}}$
   4. ${(-m^2)}^{\frac{4}{3}}$
   5. $-{\left(m^2\right)}^{\frac{4}{3}}$
   6. ${\left(3y^{\frac{4}{3}}\right)}^4$
3. Simplify as much as you can:
   $$\frac{3a^{-2}{b}^{15}{c}^{-5}}{{\left(a^{-4},b^3,c\right)}^{\frac{-5}{2}}}$$
4. Simplify as much as you can:
   $${\left(9,a^6,b^4\right)}^{\frac{1}{2}}$$
5. Simplify as much as you can:
   $${\left(a^{\frac{3}{2}},b^{\frac{3}{4}}\right)}^{16}$$
6. Simplify:
   $$x^3\sqrt{x}$$
7. Simplify:
   $$\sqrt[3]{x^4{b}^5}$$
8. Re-write the following expression as a power of $x$ :
   $$\frac{x\sqrt{x\sqrt{x\sqrt{x\sqrt{x}}}}}{\sqrt[3]{x}}$$