A mathematical refresher

Exponents

Definition of exponentiation

Multiplying the number 3 by itself four times gives

3 × 3 × 3 × 3 = 81

$3 × 3 × 3 × 3 = 81$

which can be written more simply with the shorthand notation

3^{4} = 81

$3^{4} = 81$

The notation 3 ⁴ is read as “3 raised to the fourth power” or “3 to the fourth power.” In this example, the number 3 is referred to as the base, and the number 4 is called the exponent. In the natural sciences the most frequently used base is an irrational number with the symbol e , whose value to three decimal places is 2.718. An irrational number is a number that cannot be written as a fraction. e.g., the square root of 2 ( ) and the number pi (π).

The most general representation of exponentiation is

a^{m} = c

$a^{m} = c$

where a and m can be arbitrary numbers. A few numerical examples are

\begin{matrix} 3^{4} = 81; {2.7}^{3} = 19.683; 10^{1.3} = 19.952 \dots; \\ {3.1}^{1.7} = 6.844 \dots \end{matrix}

$\begin{matrix} 3^{4} = 81; {2.7}^{3} = 19.683; 10^{1.3} = 19.952 \dots; \\ {3.1}^{1.7} = 6.844 \dots \end{matrix}$

The first two examples can be verified by hand calculation; the last two are easy to verify on a calculator.

Multiplication of exponentials

The rule for multiplying exponentials follows from the definition of exponentials:

a^{m} × a^{n} = a^{(m + n)}

$a^{m} × a^{n} = a^{(m + n)}$

This rule is easily verified by checking an example:

3^{2} × 3^{4} = (3 × 3) × (3 × 3 × 3 × 3) = 3^{6} = 3^{(2 + 4)}

$3^{2} × 3^{4} = (3 × 3) × (3 × 3 × 3 × 3) = 3^{6} = 3^{(2 + 4)}$

All the other properties of exponentials follow directly from the rule for combining exponents.

Meaning of the number 0 as exponent

We use the previous rule to deduce what a to the 0th power ( a ⁰ ) means. To make the point concrete, take a = 3. Equation AB.1 allows us to write

3^{2} × 3^{0} = 3^{(2 + 0)} = 3^{2}

$3^{2} × 3^{0} = 3^{(2 + 0)} = 3^{2}$

This expression is true if, and only if, 3 ⁰ = 1. In general, any non-0 number raised to the 0th power is equal to 1:

a^{0} = 1 (as long as a \neq 0)

$a^{0} = 1 (as long as a \neq 0)$

The reason for excluding 0 from the definition will be clear shortly.

Negative numbers as exponents

To deduce the meaning of a negative exponent, again take a = 3 as the base and use Equation AB.1 to write

3^{2} × 3^{- 2} = 3^{[2 + ( - 2)]} = 3^{0} = 1

$3^{2} × 3^{- 2} = 3^{[2 + ( - 2)]} = 3^{0} = 1$

This expression is true if, and only if, 3 ⁻² = 1/3 ² . In general,

a^{- m} = \frac{1}{a^{m}} (as long as a \neq 0)

$a^{- m} = \frac{1}{a^{m}} (as long as a \neq 0)$

Division of exponentials

The definition in Equation AB.3 extends the rule for combining exponents ( Equation AB.1 ) to include division of exponentials:

\frac{a^{m}}{a^{n}} = a^{m} × a^{- n} = a^{(m - n)}

$\frac{a^{m}}{a^{n}} = a^{m} × a^{- n} = a^{(m - n)}$

We now see why in the definition of the 0th power ( Equation AB.2 ), the base a cannot be 0: a = 0 in Equation AB.4 would force a division of 0 by 0—an operation that has no meaning.

Exponentials of exponentials

Combining exponents shows what happens when an exponential is raised to another power, for example:

{(7^{2})}^{3} = 7^{2} × 7^{2} × 7^{2} = 7^{(2 + 2 + 2)} = 7^{6}

${(7^{2})}^{3} = 7^{2} × 7^{2} × 7^{2} = 7^{(2 + 2 + 2)} = 7^{6}$

In general,

{(a^{m})}^{n} = a^{(m × n)}

${(a^{m})}^{n} = a^{(m × n)}$

Fractions as exponents

We now investigate the case in which the exponent is a fraction (i.e., a ^m ^/ ⁿ ). Taking a concrete example with a = 7, we ask what is meant by 7 ^1/2 . Applying Equation AB.1 gives

7^{1 / 2} × 7^{1/2} = 7^{(1/2 + 1/2)} = 7^{1} = 7

$7^{1 / 2} × 7^{1/2} = 7^{(1/2 + 1/2)} = 7^{1} = 7$

which immediately shows that 7 ^1/2 = . In general:

a^{\frac{1}{n}} = \sqrt[n]{a}

$a^{\frac{1}{n}} = \sqrt[n]{a}$

A direct consequence of combining Equations AB.6 and AB.4 is that

a^{\frac{m}{n}} = a^{(m × \frac{1}{n})} = {(a^{m})}^{\frac{1}{n}} = \sqrt[n]{a^{m}} = {(a^{\frac{1}{n}})}^{m} = {(\sqrt[n]{a})}^{m}

$a^{\frac{m}{n}} = a^{(m × \frac{1}{n})} = {(a^{m})}^{\frac{1}{n}} = \sqrt[n]{a^{m}} = {(a^{\frac{1}{n}})}^{m} = {(\sqrt[n]{a})}^{m}$

Because any rational decimal number can always be written as a fraction (e.g., 1.5 = 3/2, 0.47 = 47/100), decimal numbers in the exponent can be dealt with as fractions.

Equations AB.1 to AB.7 constitute the properties of exponentials that are important for computation. These properties are summarized in Box AppB.1 .

BOX AppB.1

Properties of Exponentials

\begin{matrix} a^{m} × a^{n} = a^{(m + n)} \\ \frac{a^{m}}{a^{n}} = a^{m} × a^{- n} = a^{(m - n)} \\ a^{0} = 1 (as long as a \neq 0) \end{matrix}

$\begin{matrix} a^{m} × a^{n} = a^{(m + n)} \\ \frac{a^{m}}{a^{n}} = a^{m} × a^{- n} = a^{(m - n)} \\ a^{0} = 1 (as long as a \neq 0) \end{matrix}$

\begin{matrix} a^{- m} = \frac{1}{a^{m}} (as long as a \neq 0) \\ {(a^{m})}^{n} = a^{(m × n)} \end{matrix}

$\begin{matrix} a^{- m} = \frac{1}{a^{m}} (as long as a \neq 0) \\ {(a^{m})}^{n} = a^{(m × n)} \end{matrix}$

a^{\frac{1}{n}} = \sqrt[n]{a}

$a^{\frac{1}{n}} = \sqrt[n]{a}$

a^{\frac{m}{n}} = a^{(m × \frac{1}{n})} = {(a^{m})}^{\frac{1}{n}} = \sqrt[n]{a^{m}} = {(a^{\frac{1}{n}})}^{m} = {(\sqrt[n]{a})}^{m}

$a^{\frac{m}{n}} = a^{(m × \frac{1}{n})} = {(a^{m})}^{\frac{1}{n}} = \sqrt[n]{a^{m}} = {(a^{\frac{1}{n}})}^{m} = {(\sqrt[n]{a})}^{m}$

Logarithms

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Exponents

Definition of exponentiation

Multiplication of exponentials

Meaning of the number 0 as exponent

Negative numbers as exponents

Division of exponentials

Exponentials of exponentials

Fractions as exponents

Logarithms

You're Reading a Preview

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