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Chapter 1: Exponents and surds

1.1 Revision (EMBF2)

The number system (EMBF3)

  • Discuss the number system; explain the difference between real and non-real numbers.
  • Encourage learners not to use calculators in this chapter.
  • Common misconception: \(\pi\) (irrational) ≈ \(\frac{22}{7}\) (rational).
  • Explain that the square root of a negative number is non-real.
  • Discuss raising a negative number to even and odd powers.
  • Explain that surds are a special notation or way of expressing rational exponents.
  • Key strategy in manipulation of exponential expressions: express base in terms of its prime factors.
  • Emphasize the principle of equivalence and using the additive inverse in the simplification of equations (and not “simply taking term to the other side”).
  • Rationalising the denominators is a useful tool for working with special angles in Trigonometry.
  • Learners should leave their final answers as mixed fractions.
  • Answers should always be written with positive exponents.

The diagram below shows the structure of the number system:

fadbeeea1df0ba061b4dc55668fb1215.png

We use the following definitions:

  • \(\mathbb{N}\): 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.

    Examples: \(-\frac{7}{2}; \; -\text{2,25}; \; 0; \; \sqrt{9}; \; \text{0,}\dot{8}; \; \frac{23}{1}\)

  • \(\mathbb{Q}'\): 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.

    Examples: \(\sqrt{3}; \; \sqrt[5]{2}; \; \pi; \; \frac{1 + \sqrt{5}}{2}; \; \text{1,27548}\ldots\)

  • \(\mathbb{R}\): real numbers include all rational and irrational numbers.

  • \(\mathbb{R}'\): non-real numbers or imaginary numbers are numbers that are not real.

    Examples: \(\sqrt{-25}; \; \sqrt[4]{-1}; \; -\sqrt{-\frac{1}{16}}\)

The number system

Exercise 1.1

Use the list of words below to describe each of the following numbers (in some cases multiple words will be applicable):

  • Natural (\(\mathbb{N}\))
  • Whole (\(\mathbb{N}_0\))
  • Integer (\(\mathbb{Z}\))
  • Rational (\(\mathbb{Q}\))
  • Irrational (\(\mathbb{Q}'\))
  • Real (\(\mathbb{R}\))
  • Non-real (\(\mathbb{R}'\))

\(\sqrt{7}\)

\(\mathbb{R}; \mathbb{Q}'\)

\(\text{0,01}\)

\(\mathbb{R}; \mathbb{Q}\)

\(16\frac{2}{5}\)

\(\mathbb{R}; \mathbb{Q}\)

\(\sqrt{6\frac{1}{4}}\)

\(\mathbb{R}; \mathbb{Q}\)

\(\text{0}\)

\(\mathbb{R}; \mathbb{Q}; \mathbb{Z}; \mathbb{N}_0\)

\(2\pi\)

\(\mathbb{R}' \mathbb{Q}'\)

\(-\text{5,3}\dot{8}\)

\(\mathbb{R}; \mathbb{Q}\)

\(\frac{1-\sqrt{2}}{2}\)

\(\mathbb{R}; \mathbb{Q}'\)

\(-\sqrt{-3}\)

\(\mathbb{R}'\)

\((\pi)^2\)

\(\mathbb{R}; \mathbb{Q}'\)

\(-\frac{9}{11}\)

\(\mathbb{R}; \mathbb{Q}\)

\(\sqrt[3]{-8}\)

\(\mathbb{R}; \mathbb{Q}; \mathbb{Z}\)

\(\frac{22}{7}\)

\(\mathbb{R}; \mathbb{Q}\)

\(\text{2,45897}\ldots\)

\(\mathbb{R}; \mathbb{Q}'\)

\(\text{0,}\overline{65}\)

\(\mathbb{R}; \mathbb{Q}\)

\(\sqrt[5]{-32}\)

\(\mathbb{R}; \mathbb{Q}; \mathbb{Z}\)

Laws of exponents (EMBF4)

We use exponential notation to show that a number or variable is multiplied by itself a certain number of times. The exponent, also called the index or power, indicates the number of times the multiplication is repeated.

d00a2919ce4891f6c3ec3e4f4c217e08.png
\[a^n = a \times a \times a \times \ldots \times a \quad (n \text{ times}) \qquad \left(a \in \mathbb{R}, n \in \mathbb{N}\right)\]

Examples:

  1. \(2 \times 2 \times 2 \times 2 = 2^4\)
  2. \(\text{0,71} \times \text{0,71} \times \text{0,71} = (\text{0,71})^3\)
  3. \((\text{501})^2 = \text{501} \times \text{501}\)
  4. \(k^6 = k \times k \times k \times k \times k \times k\)

For \(x^2\), we say \(x\) is squared and for \(y^3\), we say that \(y\) is cubed. In the last example we have \(k^6\); we say that \(k\) is raised to the sixth power.

We also have the following definitions for exponents. It is important to remember that we always write the final answer with a positive exponent.

  • \({a}^{0}=1\) (\(a \ne 0\) because \(0^0\) is undefined)

  • \({a}^{-n}=\frac{1}{{a}^{n}}\) (\(a \ne 0\) because \(\frac{1}{0}\) is undefined)

Examples:

  1. \(5^{-2} = \dfrac{1}{5^2} = \dfrac{1}{25}\)
  2. \((-36)^0 x = (1)x = x\)
  3. \(\dfrac{7p^{-1}}{q^{3}t^{-2}} = \dfrac{7t^2}{pq^3}\)

We use the following laws for working with exponents:

  • \({a}^{m} \times {a}^{n}={a}^{m+n}\)
  • \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}\)
  • \({\left(ab\right)}^{n}={a}^{n}{b}^{n}\)
  • \({\left(\frac{a}{b}\right)}^{n}=\frac{{a}^{n}}{{b}^{n}}\)
  • \({\left({a}^{m}\right)}^{n}={a}^{mn}\)
where \(a > 0\), \(b > 0\) and \(m, n \in \mathbb{Z}\).

Worked example 1: Laws of exponents

Simplify the following:

  1. \(5(m^{2t})^p \times 2(m^{3p})^t\)
  2. \(\dfrac{8k^3x^2}{(xk)^2}\)
  3. \(\dfrac{2^2 \times 3 \times 7^4}{(7 \times 2)^4}\)
  4. \(3(3^b)^a\)
  1. \(5(m^{2t})^p \times 2(m^{3p})^t = 10m^{2pt + 3pt} = 10m^{5pt}\)
  2. \(\dfrac{8k^3x^2}{(xk)^2} = \dfrac{8k^3x^2}{x^2k^2} = 8k^{(3-2)}x^{(2-2)} = 8k^1x^0 = 8k\)
  3. \(\dfrac{2^2 \times 3 \times 7^4}{(7 \times 2)^4} = \dfrac{2^2 \times 3 \times 7^4}{7^4 \times 2^4} = 2^{(2-4)} \times 3 \times 7^{(4-4)} = 2^{-2} \times 3 = \frac{3}{4}\)
  4. \(3(3^b)^a = 3 \times 3^{ab} = 3^{ab + 1}\)

Worked example 2: Laws of exponents

Simplify:\(\dfrac{3^m - 3^{m+1}}{4 \times 3^m - 3^m}\)

Simplify to a form that can be factorised

\[\dfrac{3^m - 3^{m+1}}{4 \times 3^m - 3^m} = \dfrac{3^m - (3^{m} \times 3)}{4 \times 3^m - 3^m}\]

Take out a common factor

\[\begin{align*} &= \dfrac{3^m(1 - 3)}{3^m(4 - 1)} \end{align*}\]

Cancel the common factor and simplify

\[\begin{align*} &= \frac{1 - 3}{4 - 1} \\ &= - \frac{2}{3} \end{align*}\]

Laws of exponents

Exercise 1.2

Simplify the following:

\(4 \times 4^{2a} \times 4^2 \times 4^a\)

\begin{align*} 4 \times 4^{2a} \times 4^2 \times 4^a &= 4^{1+2a+2+a} \\ &= 4^{3a+3} \end{align*}

\(\dfrac{3^2}{2^{-3}}\)

\begin{align*} \dfrac{3^2}{2^{-3}} &= 3^2 \times 2^3 \\ &= 9 \times 8 \\ &= 72 \end{align*}

\((3p^5)^2\)

\begin{align*} (3p^5)^2 &= 3^2 \times p^{10} \\ &= 9 p^{10} \end{align*}

\(\dfrac{k^2k^{3x-4}}{k^x}\)

\begin{align*} \dfrac{k^2k^{3x-4}}{k^x} &= \dfrac{k^{3x-2}}{k^x} \\ &= k^{3x-2-(x)} \\ &= k^{2x-2} \end{align*}

\((5^{z-1})^2+5^z\)

\[(5^{z-1})^2+5^z = 5^{2z-2}+5^z\]

\((\frac{1}{4})^0\)

\[\left ( \frac{1}{4} \right )^0 = 1\]

\((x^2)^5\)

\[\left ( x^2 \right )^5 = x^{10}\]

\(\left( \frac{a}{b} \right)^{-2}\)

\begin{align*} \left ( \frac{a}{b} \right )^{-2} &= \frac{a^{-2}}{b^{-2}} \\ &= \frac{b^2}{a^2} \end{align*}

\((m+n)^{-1}\)

\[\left ( m+n \right )^{-1} = \frac{1}{m+n}\]

\(2(p^t)^s\)

\[2\left ( p^t \right )^s = 2p^{ts}\]

\(\dfrac{1}{\left(\frac{1}{a}\right)^{-1}}\)

\[\dfrac{1}{\left ( \frac{1}{a} \right )^{-1}} = \dfrac{1}{a}\]

\(\frac{k^{0}}{k^{-1}}\)

\[\frac{k^{0}}{k^{-1}} = k\]

\(\dfrac{-2}{-2^{-a}}\)

\begin{align*} \dfrac{-2}{-2^{-a}} &= 2 \times 2^a \\ &= 2^{a+1} \end{align*}

\(\dfrac{-h}{(-h)^{-3}}\)

\begin{align*} \frac{-h}{\left ( -h \right )^{-3}} &= -h\left ( -h \right )^3 \\ &= -h\left ( -h^3 \right ) \\ &= h^4 \end{align*}

\(\left( \dfrac{a^2b^3}{c^3d} \right)^2\)

\[\left ( \frac{a^2b^3}{c^3d} \right )^2 = \frac{a^4b^6}{c^6d^2}\]

\(10^{7}(7^{0}) \times 10^{-6}(-6)^{0}-6\)

\begin{align*} 10^{7}\left ( 7^{0} \right ) \times 10^{-6}\left ( -6 \right )^{0}-6 &= 10^7(1) \times 10^{-6}(1) - 6 \\ &= 10^1 - 6 \\ &= 4 \end{align*}

\(m^3n^2 \div nm^2 \times \frac{mn}{2}\)

\begin{align*} m^3n^2 \div nm^2 \times \frac{mn}{2} &= m^3n^2 \times \frac{1}{m^2n} \times \frac{mn}{2} \\ &= \frac{m^3n^2}{m^2n} \times \frac{mn}{2} \\ &= \frac{m^2n^2}{2} \end{align*}

\((2^{-2}-5^{-1})^{-2}\)

\begin{align*} \left ( 2^{-2}-5^{-1} \right )^{-2} &= \left ( \frac{1}{4} - \frac{1}{5} \right )^{-2} \\ &= \left ( \frac{1}{20} \right )^{-2} \\ &= 20^2 \\ &= \text{400} \end{align*}

\((y^2)^{-3} \div \left( \frac{x^2}{y^3} \right)^{-1}\)

\begin{align*} \left ( y^2 \right )^{-3} \div \left ( \frac{x^2}{y^3} \right )^{-1}\div \frac{y^{-2}}{x^{-2}} &= \frac{1}{y^6} \times \frac{x^2}{y^3} \times \frac{y^{2}}{x^{2}} \\ &= \frac{1}{y^7} \end{align*}

\(\dfrac{2^{c-5}}{2^{c-8}}\)

\begin{align*} \dfrac{2^{c-5}}{2^{c-8}} &= 2^{(c-5)-(c-8)} \\ &= 2^{c-5-c+8} \\ &= 2^{3} \\ &= 8 \end{align*}

\(\dfrac{2^{9a} \times 4^{6a} \times 2^2}{8^{5a}}\)

\begin{align*} \dfrac{2^{9a} \times 4^{6a} \times 2^2}{8^{5a}} &= \frac{2^{9a} \times 2^{12a} \times 2^2}{2^{15a}} \\ &= \frac{2^{9a+12a+2}}{2^{15a}} \\ &= 2^{21a+2-15a} \\ &= 2^{6a+2} \end{align*}

\(\dfrac{20t^5p^{10}}{10t^4p^9}\)

\begin{align*} \dfrac{20t^5p^{10}}{10t^4p^9} &= 2t^{5-4}p^{10-9} \\ &= 2pt \end{align*}

\(\left( \dfrac{9q^{-2s}}{q^{-3s}y^{-4a-1}} \right)^2\)

\begin{align*} \left( \dfrac{9q^{-2s}}{q^{-3s}y^{-4a-1}} \right)^2 &= \frac{\left( 9q^{-2s} \right)^2}{\left( q^{-3s}y^{-4a-1} \right)^2} \\ &= \frac{ 81q^{-4s} }{ q^{-6s}y^{-8a-2} } \\ &= \frac{ 81q^{6s} }{ q^{4s}y^{-(8a+2)} } \\ &= 81q^{2s} y^{8a+2} \end{align*}