In this chapter, you will revise work on exponents that you have done in previous grades.You will extend the laws of exponents to include exponents that are negative numbers.

You will also solve simple equations in exponential form.

In Grade 8 you learnt about scientific notation.In this chapter we will extend the scientific notation to include very small numbers such as 0,0000123.

Revision

Remember that exponents are a shorthand way of writing repeated multiplication of the same number by itself. For example:\( 5 \times 5 \times 5 = 5^3\). The exponent, which is 3 in this example, stands for however many times the value is being multiplied. The number that is being multiplied, which is 5 in this example, is called the base.

If there are mixed operations, then the powers should be calculated before multi-plication and division. For example: \(5^2 \times 3^2 = 25 \times 9\).

You learnt these laws for working with exponents in previous grades:

Law

Example

\(a^m \times a^n = a^{m+n}\)

\(3^2\times 3^3 = 3^{2+3}=3^5\)

\(a^m \div a^n =a^{m-n}\)

\(5^4 \div 5^2 = 5^{4-2} =5^2\)

\((a^m)^n = a^{m+n}\)

\((2^3)^2 = 2^{2 \times 3} =2^6\)

\((a \times t)^n = a^n \times t^n\)

\((3 \times 4)^2 = 3^2 \times 4^2\)

\(a^0 = 1\)

\(32^0 =1\)

The exponential form of a number

  1. Write the following in exponential notation:

    1. \( 2 \times 2 \times 2 \times 2 \times 2 \)

    2. \(s \times s \times s \times s \)

    3. \( (-6) \times (-6) \times (-6)\)

    4. \( 2\times 2 \times 2 \times 2 \times s\times s \times s \times s\)

    5. \( 3 \times 3 \times 3 \times 7 \times 7\)

    6. \( 500 \times (1,02) \times (1,02)\)

  2. Write each of the numbers in exponential notation in some different ways if possible:

    1. 81

    2. 125

    3. 1 000

    4. 64

    5. 216

    6. 1 024

Order of operations

  1. Calculate the value of \(7^2 - 4\).

    Bathabile did the calculation like this:\( 7^2 - 4 = 14 - 4 = 10\)

    Nathaniel did the calculation differently:\( 7^2 - 4 = 49 - 4 = 45\)

    Which learner did the calculation correctly? Give reasons for your answer.

  2. Calculate: \(5 + 3 \times 2^2 - 10\), with explanations.

  3. Explain how to calculate \( 2^6 - 6^2\).

  4. Explain how to calculate \( (4 + 1)^2 + 8 \times \sqrt[3]{64} \)

Laws of exponents

  1. Use the laws of exponents to calculate the following:

    1. \( 2^2 \times 2^4\)

    2. \( 3^4 \div 3^2\)

    3. \( 3^0 + 3^4\)

    4. \((2^3)^2\)

    5. \((2 \times 5)^2\)

    6. \( (2^2 \times 7)^3\)

  2. Complete the table. Substitute the given number for y. The first column has been done as an example.

    y

    2

    3

    4

    5

    (a)

    \(y \times y^4\)

    \(2 \times 2 ^4\)

    \(= 2^{1+ 4}\)

    \(= 2^5\)

    \(= 32\)

    (b)

    \(y^2 \times y^3\)

    \(2^2 \times 2^3\)

    \(= 2 ^{2+ 3}\)

    \(= 4 \times 8\)

    \(= 32\)

    (c)

    \(y^5\)

    \(2^5 = 32\)

  3. Are the expressions \(y \times y^4; y ^2 \times y^3 \) and \(y^5\) equivalent? Explain.

  4. Complete the table. Substitute the given number for y.

    y

    2

    3

    4

    5

    (a)

    \(y^4 \div y^2\)

    \(2^4 \div 2^2\)

    \(= 16 \div 4\)

    \(= 4\)

    (b)

    \(y^3 \div y^1 \)

    \(2^3 \div 2^1\)

    \(= 8 \div 2\)

    \(= 4\)

    (c)

    \(y^2\)

    \(2^{2}\) = 4

    1. From the table, is \( y^4 \div y^2 = y^3 \div y^1 = y^2 \)? Explain why.

    2. Evaluate \(y^4 \div y^2 \) for \(y = 15\).

  6. Complete the table:

    \(x\)

    2

    3

    4

    5

    (a)

    \(2 \times 5^x\)

    \(2 \times 5^2\)

    \(= 2 \times 25\)

    \(= 50\)

    (b)

    \((2 \times 5)^x\)

    \((2 \times 5)^2\)

    \(= 10^2\)

    \(= 100\)

    (c)

    \( 2^x \times 5^x\)

    \(2^2 \times 5^2\)

    \(= 4 \times 25\)

    \(= 100\)

    1. From the table above, is \(2 \times 5^x =(2 \times 5)^x\) ? Explain.

    2. Which expressions in question 6 are equivalent? Explain.

  8. Below is a calculation that Wilson did as homework. Mark each problem correct or incorrect and explain the mistakes.

    1. \( b^3 \times b^ 8= b^{24}\)

    2. \((5x)^2 = 5x^2\)

    3. \((-6a) \times (-6a) \times (-6a) = (-6a)^3\)

Integer exponents

\(5^4\) means \(5 \times 5 \times 5 \times 5\). The exponent 4 indicates the number of appearances of the repeated factor.

What may a negative exponent mean, for example what may \(5^{-4}\) mean?

Mathematicians have decided to use negative exponents to indicate repetition of the multiplicative inverse of the base, for example \(5^{-4}\) is used to indicate \(\frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} \) or \((\frac{1}{5})^4 \), and \(x^{-3}\) is used to indicate \((\frac{1}{x})^3 \) which is \(\frac{1}{x} \times \frac{1}{x} \times \frac{1}{x} \)

This decision was not taken blindly - mathematicians were well aware that it makes good sense to use negative exponents in this meaning. One major advantage is that the negative exponents, when used in this meaning, have the same properties as positive exponents, for example:

\(2^{-3} \times 2^{-4} = 2^{(-3)+(-4)} =2^{-7} \)because \(2^{-3} \times 2^{-4}\) means \((\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} )\times (\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} \times \frac{1}{2} )\) which is \((\frac{1}{2})^7\).

\(2^{-3} \times 2^4 = 2^{(-3)+4} = 2 ^1 \) because \(2^{-3} \times 2^4\) means \((\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}) \times (2 \times 2 \times 2 \times 2)\) which is 2.

Negative exponents

  1. Express each of the following in the exponential notation in two ways: with positive exponents and with negative exponents.

    1. \(\frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5}\)

    2. \(\frac{1}{3} \times\frac{1}{3} \times\frac{1}{3} \times\frac{1}{3}\)

  2. In each case, check whether the statement is true or false. If it is false, write a correct statement. If it is true, give reasons why you say so.

    1. \(10^{-3}= 0,001\)

    2. \( 3 ^{-5} \times 9^2 =3\)

    3. \( 25 ^2 \times 10 ^{-6} \times 2 ^6 = 5 \)

    4. \( (\frac{1}{5})^{-4} = 5 ^4 \)

  3. Calculate each of the following without using a calculator:

    1. \( 10^{-3} \times 20^4\)

    2. \( (\frac{1}{5})^{-4} \)

    1. Use a scientific calculator to determine the decimal values of the given powers.

      Example: To find \(3^{-1} \) on your calculator, use the key sequence: \(3 y^x 1± = \)

      Power

      \(2^ {-1}\)

      \(5^ {-1}\)

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

      \((0,3)^{-1}\)

      \(0^{-1}\)

      \(10^{-1}\)

      \(10^{-2}\)

      Decimal value

    2. Explain the meaning of \(10^{-3}\).

  5. Determine the value of each of the following in two ways:

    A. By using the definition of powers (For example, \(5^2 \times 5^ 0 = 25 \times 1 = 25\).)

    B. By using the properties of exponents (For example, \(5^2 \times 5^0 = 5^{2 + 0} = 5^2 = 25\).)

    1. \((3^3)^{-2}\)

    2. \(4^2 \times 4^{-2}\)

    3. \(5^{-2} \times 5^{-1}\)

  6. Calculate the value of each of the following. Express your answers as common fractions.

    1. \(2^{-3}\)

    2. \(3^2 \times 3^{-2}\)

    3. \((2 +3)^{-2}\)

    4. \(3^{-2} \times 2^{-3}\)

    5. \(2^{-3} + 3^{-3}\)

    6. \(10^{-3}\)

    7. \(2^3 + 2^{-3}\)

    8. \((3^{-1})^{-1}\)

    9. \((2^{-3})^2\)

  7. Which of the following are true? Correct any false statement.

    1. \( 6 ^{-1}= -6\)

    2. \(3x^{-2} = \frac{1}{3x^2}\)

    3. \(3^{-1}x^{-2} = \frac{1}{3x^2}\)

    4. \((ab)^{-2} = \frac{1}{a^2 b^2}\)

    5. \((\frac{2}{3})^{-2} = (\frac{3}{2})^2\)

    6. \((\frac{1}{3})^{-1} = 3 \)

Solving simple exponential equations

An exponential equation is an equation in which the variable is in the exponent. So when you solve exponential equations, you are solving questions of the form "To what power must the base be raised for the statement to be true?"

To solve this kind of equation, remember that:

If \(a^m = a^n\), then \( m = n\).

In other words, if the base is the same on either side of the equation, then the exponents are the same.

Example:

\(3^x = 243\)

\(3^x = 3^5\) (rewrite using the same base)

\(x = 5\) (since the bases are the same, we equate the exponents)

Some exponential equations are slightly more complex:

Example:

\(3^{x+3} = 243\)

\(3^{x+3} = 3^5 \)(rewrite using the same base)

\(x + 3 = 5\) (equate the exponents)

x \(= 2\)

Check:

LHS \( 3^{2+3}=3^5 = 243\)

Remember that the exponent can also be negative. However, you follow the same method to solve these kinds of equations.

Example:

\(2^x=\frac{1}{32}\)

\(2^x =2^{-5} \) (rewrite using the same base)

\(x = -5\) (equate the exponents)

Solving exponential equations

  1. Use the table to answer questions that follow:

    x

    2

    3

    4

    5

    \(2^x\)

    4

    8

    16

    32

    \(3^x\)

    9

    27

    81

    243

    \(5^x\)

    25

    125

    625

    3 125

    For which value of x is:

    1. \( 2^x = 32\)

    2. \(3^x = 81\)

    3. \(5^x = 3125\)

    4. \(2^x =8\)

    5. \(5^x =625\)

    6. \(3^x = 9\)

    7. \(5^{x+1} = 25\)

    8. \(3^{x+2}=27\)

    9. \(2^{x-1} = 8 \)

  2. Solve these exponential equations. You may use your calculator if necessary.

    1. \(4^x = \frac{1}{64}\)

    2. \(6^{2x}=1 296\)

    3. \(2^{x-1} = \frac{1}{8}\)

    4. \(3^{x+2}= \frac{1}{729}\)

    5. \(5^{x+1}= 15 625\)

    6. \(2^{x+3}=\frac{1}{4}\)

    7. \(4^{x+3}=\frac{1}{256}\)

    8. \(3^{2-x} =81\)

    9. \(5^{3x}= \frac{1}{125}\)

Scientific notation

Scientific notation is a way of writing numbers that are too big or too small to be written clearly in decimal form. The diameter of a hydrogen atom, for example, is a very small number. It is 0,000000053 mm. The distance from the sun to the earth is, on average, 150 000 000 km.

In scientific notation the diameter of the hydrogen molecule is written as \( 5,3 \times 10^{-8}\) and the distance from the sun to the earth as \(1,5 \times 10^{8}\). It is easier to compare and to calculate numbers like these, as it is very cumbersome to count the zeros when you work with these numbers.

Look at more examples below:

Decimal notation

Scientific notation

\(6~130~000\)

\(6,13 \times 10^6\)

\(0,00001234\)

\(1,234 \times 10^{-5}\)

A number written in scientific notation is written as the product of two numbers, in the form ± \(a \times 10^n\) where a is a decimal number between 1 and 10, and n is an integer.

Any number can be written in scientific notation, for example:

\(40 = 4,0 \times 10\)

\(2 = 2 \times 10^0\)

The decimal number 324 000 000 is written as \(3,24 \times 10 ^8\) in scientific notation, because the decimal comma is moved 8 places to the left to form 3,24.

The decimal number 0,00000065 written in scientific notation is \(6,5 \times 10^{-7}\), because the decimal point is moved 7 places to the right to form the number 6,5.

Writing very small and very large numbers

  1. Express the following numbers in scientific notation:
    1. 134,56
    2. 0,0000005678
    3. 876 500 000
    4. 0,0000000000321
    5. 0,006789
    6. 89 100 000 000 000
    7. 0,001
    8. 100

  2. Express the following numbers in ordinary decimal notation:

    1. \(1,234 \times 10^6\)

    2. \(5 \times 10^{-1}\)

    3. \( 4,5 \times 10^5\)

    4. \( 6,543 \times 10^{-11}\)

  3. Why do we say that \(34 \times 10^3\) is not written in scientific notation? Rewrite it in scientific notation.

  4. Is each of these numbers written in scientific notation? If not, rewrite it so that it is in scientific notation.

    1. \( 90,3 \times 10^{-5}\)

    2. \(100 \times 10^2\)

    3. \(1,36\times 10^5\)

    4. \(2,01 \times 10^{-2}\)

    5. \(0,01 \times 10^3\)

    6. \(0,6 \times 10^8\)

Calculations using scientific notation

Example:

\( 123 000 \times 4 560 000\)

\(= 1,23 \times 10^{5} \times 4, 56 \times 10^6\) (write in scientific notation)

\(= 1,23 \times 4,56 \times 10^5 \times 10^6\) (multiplication is commutative)

\(= 5,6088 \times 10^{11}\)(Use a calculator to multiply the decimals, but add the powers mentally.)

  1. Use scientific notation to calculate each of the following. Give the answer in scientific notation.

    1. \( 135 000 \times 246 000 000\)

    2. \( 987 654 \times 123 456\)

    3. \( 0,000065 \times 0,000216 \)

    4. \( 0,000000639 \times 0,0000587\)

    Example:

    \( 5 \times 10^3 + 4 \times 10^4\)

    \(= 0,5 \times 10^4 + 4 \times 10^4\)(Form like terms)

    \(= 4,5 \times 10^4\)(Combine like terms)

  2. Calculate the following. Leave the answer in scientific notation.
    1. \( 7,16 \times 10^5 +2,3 \times 10^3\)
    2. \(2,3 \times 10^{-4} + 6,5 \times 10^{-3}\)
    3. \( 4,31 \times 10^7 + 1,57 \times 10^6\)
    4. \( 6,13 \times 10^{-10} + 3,89 \times 10^{-8}\)