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Working with numbers in exponential form

3.2 Working with numbers in exponential form

Whole numbers

When writing exponents, we use a superscript to show how many times a number is multiplied by itself. Look at this example:

\[2^{\mathbf{5}} = 2 \times 2 \times 2 \times 2 \times 2\]

The number \(2\) is multiplied by itself \(\mathbf{5}\) times. We read this as “\(2\) to the power of \(\mathbf{5}\)”. The number \(2\) is the base, and the number \(\mathbf{5}\) is the exponent. When we write \(2 \times 2 \times 2 \times 2 \times 2\) as \(2^{5}\), we are using exponential notation or exponential form.

exponential form
an expression written with an exponent, where the bottom part is the base and the superscript part is the exponent or index
Exercise 3.1: Writing repeated multiplication in exponential form
Use your calculator to check these answers.
  1. \[2 \times 2 \times 2 =\]
  2. \[2 \times 2 \times 2 \times 2 \times 2 \times 2 =\]
  3. \[3 \times 3 \times 3 =\]
  4. \[3 \times 3 \times 3 \times 3 \times 3 \times 3 =\]

Write each of those products in exponential form.

  1. \[\begin{align} &2 \times 2 \times 2 = 8 \\ &2^{3} = 8 &&\text{The number 2 is multiplied by itself 3 times, so 3 is the exponent.} \end{align}\]
  2. \[\begin{align} &2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64 \\ &2^{6} = 64 &&\text{The number 2 is multiplied by itself 6 times, so 6 is the exponent.} \end{align}\]
  3. \[\begin{align} &3 \times 3 \times 3 = 27 \\ &3^{3} = 27 &&\text{The number 3 is multiplied by itself 3 times, so 3 is the exponent.} \end{align}\]
  4. \[\begin{align} &3 \times 3 \times 3 \times 3 \times 3 \times 3 = 729 \\ &3^{6} = 729 &&\text{The number 3 is multiplied by itself 6 times, so 6 is the exponent.} \end{align}\]

Instead of writing \(3 \times 3 \times 3 \times 3 \times 3 \times 3\), in exponential form we write \(3^{6}\). You will get the same answer using a calculator, regardless of the form.

Exercise 3.2: Writing exponents as products of bases

Write each power as a product of the same base. Calculate the answer.

  1. \[5^{2}\]
  2. \[2^{5}\]
  3. \[3^{4}\]
  4. \[4^{3}\]
  5. \[2^{7}\]
  6. \[3^{5}\]
  7. \[10^{2}\]
  8. \[15^{2}\]

We look at the exponent to decide how many times to multiply the base by itself.

  1. \[\begin{align*} &5^{\mathbf{2}} = 5 \times 5 = 25 &&\mathbf{2 \ times} \end{align*}\]
  2. \[\begin{align*} &2^{\mathbf{5}} = 2 \times 2 \times 2 \times 2 \times 2 = 32 &&\mathbf{5 \ times} \end{align*}\]
  3. \[\begin{align*} &3^{\mathbf{4}} = 3 \times 3 \times 3 \times 3 = 81 &&\mathbf{4 \ times} \end{align*}\]
  4. \[\begin{align*} &4^{\mathbf{3}} = 4 \times 4 \times 4 = 64 &&\mathbf{3 \ times} \end{align*}\]
  5. \[\begin{align*} &2^{\mathbf{7}} = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128 &&\mathbf{7 \ times} \end{align*}\]
  6. \[\begin{align*} &3^{\mathbf{5}} = 3 \times 3 \times 3 \times 3 \times 3 = 243 &&\mathbf{5 \ times} \end{align*}\]
  7. \[\begin{align*} &10^{\mathbf{2}} = 10 \times 10 = 100 &&\mathbf{2 \ times} \end{align*}\]
  8. \[\begin{align*} &15^{\mathbf{2}} = 15 \times 15 = 225 &&\mathbf{2 \ times} \end{align*}\]

Compare exponents using inequalities

When we need to compare two numbers written in exponential form, it is sometimes easier to first work out the answer for each.

Worked example 3.1: Comparing exponents

Compare the following. Use the \(<\), \(>\) or \(=\) sign between the two numbers.

  1. \[3^{2}\ ☐ \ 5^{2}\]
  2. \[2^{2}\ ☐ \ 3^{2}\]
  3. \[6^{2}\ ☐ \ 4^{2}\]
  4. \[7^{2}\ ☐ \ 6^{2}\]

Work out the answer on each side.

  1. \[\begin{align*} 3^{2}\ ☐ \ 5^{2} \\ 9\ ☐ \ 25 \end{align*}\]
  2. \[\begin{align*} 2^{2}\ ☐ \ 3^{2} \\ 4\ ☐ \ 9 \end{align*}\]
  3. \[\begin{align*} 6^{2}\ ☐ \ 4^{2} \\ 36\ ☐ \ 16 \end{align*}\]
  4. \[\begin{align*} 7^{2}\ ☐ \ 6^{2} \\ 49\ ☐ \ 36 \end{align*}\]

Compare the numbers and place the correct sign between the two numbers.

  1. \(9 < 25\), so \(3^{2} < 5^{2}\)
  2. \(4 < 9\), so \(2^{2} < 3^{2}\)
  3. \(36 > 16\), so \(6^{2} > 4^{2}\)
  4. \(49 > 36\), so \(7^{2} > 6^{2}\)

When the bases are different and the powers are the same, we can compare the exponents by comparing only their bases.

So, given \(\mathbf{3}^{2}\) and \(\mathbf{5}^{2}\), we can see that \(\mathbf{3}^{2} < \mathbf{5}^{2}\) because \(\mathbf{3} < \mathbf{5}\).

Worked example 3.2: Comparing exponents

Use the correct symbol (\(<\), \(>\), \(=\)) to make these statements true.

  1. \[3^{3}\ ☐ \ 3^{2}\]
  2. \[2^{4}\ ☐ \ 2^{5}\]
  3. \[6^{3}\ ☐ \ 6^{4}\]
  4. \[7^{2}\ ☐ \ 7^{2}\]

The statement is true if the correct inequality sign is placed between the two exponents. Calculate the answer on each side.

  1. \[\begin{align*} &3^{3}\ ☐ \ 3^{2} \\ &27 > 9 \text{, so } 3^{3} > 3^{2} \end{align*}\]
  2. \[\begin{align*} &2^{4}\ ☐ \ 2^{5} \\ &16 < 32 \text{, so } 2^{4} < 2^{5} \end{align*}\]
  3. \[\begin{align*} &6^{3}\ ☐ \ 6^{4} \\ &216 < 1\ 296 \text{, so } 6^{3} < 6^{4} \end{align*}\]
  4. \[\begin{align*} &7^{2}\ ☐ \ 7^{2} \\ &49 = 49 \text{, so } 7^{2} = 7^{2} \end{align*}\]

When the bases are the same and the powers are different, we can compare the exponents by comparing only the powers.

So, given \(3^{\mathbf{3}}\) and \(3^{\mathbf{2}}\), we can see that \(3^{\mathbf{3}} > 3^{\mathbf{2}}\) because \(\mathbf{3} > \mathbf{2}\).

Worked example 3.3: Comparing exponents

Use the correct symbol (\(<\), \(>\), \(=\)) to make these statements true.

  1. \[3^{3}\ ☐ \ 4^{2}\]
  2. \[2^{4}\ ☐ \ 3^{5}\]
  3. \[3^{5}\ ☐ \ 4^{3}\]
  4. \[6^{2}\ ☐ \ 2^{6}\]

Calculate the answers on each side.

  1. \[\begin{align*} &3^{3}\ ☐ \ 4^{2} \\ &27 > 16 \text{, so } 3^{3} > 4^{2} \end{align*}\]
  2. \[\begin{align*} &2^{4}\ ☐ \ 3^{5} \\ &16 < 243 \text{, so } 2^{4} < 3^{5} \end{align*}\]

    In this question, both the base and the power on the left are smaller than the base and the power on the right. So, the answer was easy to find without doing any calculations.

  3. \[\begin{align*} &3^{5}\ ☐ \ 4^{3} \\ &243 > 64 \text{, so } 3^{5} > 4^{3} \end{align*}\]
  4. \[\begin{align*} &6^{2}\ ☐ \ 2^{6} \\ &36 = 36 \text{, so } 6^{2} = 2^{6} \end{align*}\]
temp text

Integers in exponential form

When multiplying a positive (or negative) integer by a negative (or positive) integer, we need to be careful to see when the sign changes from \(−\) to \(+\) or from \(+\) to \(−\). For example, \(- 2 \times ( - 2) = + 4\), but \(- 2 \times 2 = - 4\).

Remember, when multiplying or dividing two negative integers, the result is a positive number. If the signs of the two integers differ, then the result of the multiplication or division will be a negative number.

Exercise 3.3: Write repeated multiplication of integers in exponential form

Calculate the following without using a calculator.

Hint: Be careful about the sign changes at each step. Group two numbers together to avoid making mistakes.

\[- 2 \times (- 2) \times (- 2) =\]
\[- 2 \times (- 2) \times (- 2) = ( - 2 \times - 2) \times ( - 2) = 4 \times ( - 2) = - 8\]
\[- 2 \times (- 2) \times (- 2) \times (- 2) =\]
\[- 2 \times (- 2) \times (- 2) \times (- 2) = ( - 2 \times (- 2)) \times ( - 2 \times (- 2)) = 4 \times 4 = 16\]
\[- 5 \times (- 5) =\]
\[- 5 \times (- 5) = 25\]
\[- 5 \times (- 5) \times (- 5) =\]
\[- 5\ \times (- 5) \times (- 5) = ( - 5\ \times (- 5)) \times ( - 5) = 25 \times ( - 5) = - 125\]
\[- 1 \times (- 1) \times (- 1) =\]
\[- 1\ \times \ (- 1)\ \times \ (- 1)\ = \ ( - 1\ \times \ (- 1))\ \times \ ( - 1)\ = \ 1\ \times \ ( - 1)\ = \ - 1\]
\(- 1 \times (- 1) \times (- 1) \times (- 1) =\)
\[- 1\ \times \ (- 1)\ \times \ (- 1)\ \times \ (- 1)\ = \ ( - 1\ \times \ (- 1))\ \times \ ( - 1\ \times \ (- 1))\ = \ 1\ \times \ 1\ = \ 1\]

Worked example 3.4: Write repeated multiplication of integers in exponential form

Calculate the following. Compare the answers.

\(-2^2\) and \((-2)^2\)

These numbers are written in exponential form. The minus sign is not always inside the bracket, which means that it doesn’t always belong to the power.

Calculate the first answer.

\(-2^{2}\): the minus is not part of the power, so \(- 2^{2} = - \left( 2^{2} \right) = - (2 \times 2) = - 4\)

Calculate the second answer.

\((-2)^{2}\): the minus is part of the power, so \(( - 2)^{2} = ( - 2) \times ( - 2) = 4\)

Compare the answers.

The two answers differ in sign: the first one is negative and second one is positive.

Use your calculator to calculate the answers.

If your answers are different from those of the calculator, try to explain how the calculator did the calculations differently.

The calculator “understands” \(- 5^{2}\) and \(( - 5)^{2}\) as two different calculations.

It understands \(-5^{2}\) as \(-5 \times 5 = -25\) and \(( -5)^{2}\) as \(-5 \times -5 = 25\).

Worked example 3.5: Write repeated multiplication of integers in exponential form

Calculate the following. Compare the answers.

\((-5)^3\) and \(-5^3\)

Calculate the first answer.

\(( - 5)^{3}\): the minus is part of the power, so \(( - 5)^{3} = ( - 5) \times ( - 5) \times ( - 5) = 25 \times ( - 5) = - 125\)

Calculate the second answer.

\(- 5^{3}\): the minus is not part of the power, so \(- 5^{3} = - (5 \times 5 \times 5) = - (25 \times 5) = - 125\)

Compare the answers.

The two answers are the same, both are negative.

When the base is a negative number and the exponent is an even number, the answer is positive: \(( - 5)^{2} = 25\)

When the base is a negative number and the exponent is an odd number, the answer is negative: \(( - 5)^{3} = - 125\)

Exercise 3.4: Exponents with negative numbers

Say whether the sign of the answer is negative or positive. Explain why.

  1. \[( - 3)^{6}\]
  2. \[( - 5)^{11}\]
  3. \[( - 4)^{20}\]
  4. \[( - 7)^{5}\]

All these questions have a negative base, so we need to look at whether the exponent is an even or odd number.

  1. \((-3)^{6}\) The answer will be positive, because \(6\) is an even number.
  2. \((-5)^{11}\) The answer will be negative, because \(11\) is an odd number.
  3. \((-4)^{20}\) The answer will be positive, because \(20\) is an even number.
  4. \((-7)^{5}\) The answer will be negative, because \(5\) is an odd number.

Say whether the following statements are true or false. If a statement is false, rewrite it as a correct statement.

  1. \[( - 3)^{2} = - 9\]
  2. \[- 3^{2} = 9\]
  3. \[( - 5^{2}) = 5^{2}\]
  4. \[( - 1)^{3} = - 1^{3}\]
  5. \[( - 6)^{3} = - 18\]
  6. \[( - 2)^{6} = 2^{6}\]

If the statement is true, then the LHS = RHS in each equation.

  1. \[\begin{align*} &( - 3)^{2} = - 9 &&\text{False. } ( - 3)^{2} = 9 \end{align*}\]
  2. \[\begin{align*} &- 3^{2} = 9 &&\text{False. } - 3^{2} = - 9 \end{align*}\]
  3. \[\begin{align*} &( - 5^{2}) = 5^{2} &&\text{False. } ( - 5^{2}) = ( - 25)\ = - 25 \end{align*}\]
  4. \[\begin{align*} &( - 1)^{3} = - 1^{3} &&\text{True.} \end{align*}\] \[\begin{align*} &&\text{The exponent is an odd number, so the answer will be negative on both sides.} \end{align*}\]
  5. \[\begin{align*} &( - 6)^{3} = - 18 &&\text{False. } ( - 6)^{3} = - 216 \end{align*}\]
  6. \[\begin{align*} &( - 2)^{6} = 2^{6} &&\text{True.} \end{align*}\] \[\begin{align*} &&\text{The exponent is an even number, so the answer will be positive on both sides.} \end{align*}\]

Rational numbers in exponential form

Common fractions and decimal fractions are examples of rational numbers.

rational number
a number that can be written as a ratio of two integers

Decimal fractions need to have a limited number of decimal places – or have a repeating pattern after the comma – to be considered a rational number.

Let’s look at fractions and decimals in exponential form. Squaring or cubing a fraction or a decimal fraction is no different from squaring or cubing an integer.

Exercise 3.5: Find squares of fractions and decimals

Find the value of the square of the following fractions.

  1. \[\frac{1}{2}\]
  2. \[\frac{1}{4}\]
  3. \[\frac{2}{3}\]
  4. \[\frac{4}{7}\]

To find the value of the square of the fraction, we begin by multiplying the given fraction by itself.

  1. \[\frac{1}{2} \times \frac{1}{2} = \frac{1 \times 1}{2 \times 2} = \frac{1}{4}\]
  2. \[\ \frac{1}{4} \times \frac{1}{4} = \frac{1 \times 1}{4 \times 4} = \frac{1}{16}\]
  3. \[\frac{2}{3} \times \frac{2}{3} = \frac{2 \times 2}{3 \times 3} = \frac{4}{9}\]
  4. \[\frac{4}{7} \times \frac{4}{7} = \frac{4 \times 4}{7 \times 7} = \frac{16}{49}\]

    Do you see the pattern? To square a fraction, you square the numerator and the denominator. So, \(\left( \frac{1}{2} \right)^{2} = \frac{1^{2}}{2^{2}} = \frac{1}{4}\).

Calculate the following.

  1. \[\left( \frac{2}{7} \right)^{2}\]
  2. \[\left( \frac{5}{3} \right)^{2}\]
  3. \[\left( \frac{3}{11} \right)^{2}\]

Now we can use the rule found in Question 1 to complete these squares of fractions.

  1. \[\left(\frac{2}{7}\right)^{2} = \frac{2^{2}}{7^{2}} = \frac{4}{49}\]
  2. \[\left(\frac{5}{3} \right)^{2} = \frac{5^{2}}{3^{2}} = \frac{25}{9}\]
  3. \[\left(\frac{3}{11} \right)^{2} = \frac{3^{2}}{11^{2}} = \frac{9}{121}\]

Do these calculations.

  1. Use the fact that \(\text{0,6}\) can be written as \(\frac{6}{10}\) to calculate its square.
  2. Use the fact that \(\text{0,8}\) can be written as \(\frac{8}{10}\) to calculate its square.
  1. We know that \(\text{0,6}\) can be written as \(\frac{6}{10}\) and we know how to square a fraction.

So, \((\text{0,6})^{2} = \left( \frac{6}{10} \right)^{2} = \frac{6^{2}}{10^{2}} = \frac{36}{100} = \text{0,36}\).

  1. We know that \(\text{0,8}\) can be written as \(\frac{8}{10}\) and we know how to square a fraction.

So, \((\text{0,8})^{2} = \left( \frac{8}{10} \right)^{2} = \frac{8^{2}}{10^{2}} = \frac{64}{100}\).

Do you see the pattern? To square a decimal, you square the number (ignoring the comma) and then double the number of digits after the comma.

Step 1: \(6^{2} = \mathbf{36}\)

Step 2: There is one digit after the comma in \(\text{0,6}\), so the answer will have two digits to the right of the decimal comma: \(0\),\(\mathbf{36}\).

So, \((0,6)^{2} = \text{0,36}\).

Exercise 3.6: Find cubes of fractions and decimals

Calculate the following cubes of fractions.

  1. \[\left( \frac{1}{2} \right)^{3}\]
  2. \[\left( \frac{2}{5} \right)^{3}\]
  3. \[\left( \frac{3}{5} \right)^{3}\]
  4. \[\left( \frac{3}{10} \right)^{3}\]

We can use the same strategy as we used with squares to calculate cubes of fractions.

  1. \[\left( \frac{1}{2} \right)^{3} = \frac{1^{3}}{2^{3}} = \frac{1}{8}\]
  2. \[\left( \frac{2}{5} \right)^{3} = \frac{2^{3}}{5^{3}} = \frac{8}{125}\]
  3. \[\left( \frac{3}{5} \right)^{3} = \frac{3^{3}}{5^{3}} = \frac{27}{125}\]
  4. \[\left( \frac{3}{10} \right)^{3} = \frac{3^{3}}{10^{3}} = \frac{27}{1\ 000}\]

Another way of writing \(\frac{27}{1\ 000}\) is \(\text{0,027}\).

  1. Use the fact that \(\text{0,6}\) can be written as \(\frac{6}{10}\) to calculate its cube.
  2. Use the fact that \(\text{0,8}\) can be written as \(\frac{8}{10}\) to calculate its cube.
  1. We know that \(\text{0,6} = \frac{6}{10}\)
    So, \((\text{0,6})^{3} = \left( \frac{6}{10} \right)^{3} = \frac{216}{1\ 000} = \text{0,216}\).

  2. We know that \(\text{0,8}\ = \ \frac{8}{10}\)
    So, \((\text{0,8})^{3} = \left( \frac{8}{10} \right)^{3} = \frac{512}{1\ 000} = \text{0,512}\).

Scientific notation

The scientific notation of very large numbers makes calculations easier.

For example, we can write \(136\ 000\ 000\) as \(\text{1,36} \times 100\ 000\ 000 = \text{1,36} \times 10^{8}\).

\(\text{1,36} \times 10^{8}\) is called the scientific notation for \(136\ 000\ 000\).

In scientific notation, a number is expressed in two parts: a number between \(1\) and \(10\) multiplied by a power of \(10\). The exponent must always be an integer.

Exercise 3.7: Writing big numbers in scientific notation

Write the following numbers in scientific notation.

  1. \[367\ 000\ 000\]
  2. \[21\ 900\ 000\]
  3. \[600\ 000\ 000\ 000\]
  4. \[178\ 000\]

Remember, the first part of the number must be between \(1\) and \(10\).

  1. \[3 \mathbf{67\ 000\ 000} = \text{3,67} \times 10^{\mathbf{8}}\]
  2. \[2 \mathbf{1\ 900\ 000} = \text{2,19} \times 10^{\mathbf{7}}\]
  3. \[6 \mathbf{00\ 000\ 000\ 000} = 6 \times 10^{\mathbf{11}}\]
  4. \[1 \mathbf{78\ 000} = \text{1,78} \times 10^{\mathbf{5}}\]

Write each of the following numbers in the ordinary way.

  1. \[\text{1,24} \times 10^{5}\]
  2. \[\text{9,2074} \times 10^{6}\]
  3. \[\text{1,04} \times 10^{6}\]
  4. \[\text{2,05} \times 10^{3}\]

To write the number given in scientific notation in the ordinary way, we need to multiply by the power of \(10\).

  1. \[\text{1,24} \times 10^{5} = \text{1,24} \times 100\ 000 = 124\ 000\]
  2. \[\text{9,2074} \times 10^{6} = \text{9,2074} \times 1\ 000\ 000 = 9\ 207\ 400\]
  3. \[\text{1,04} \times 10^{6} = \text{1,04} \times 1\ 000\ 000 = 1\ 040\ 000\]
  4. \[\text{2,05} \times 10^{3} = \text{2,05} \times 1\ 000 = 2\ 050\]