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AI SL · 12.6

Exponents & Logarithms

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12.6.1

Laws of exponents: negative and fractional indices

12.6.1

Laws of exponents: negative and fractional indices

Target: I can use the laws of exponents with integer powers, including negative powers, and write a power as a fraction. (+2 more)

By the end of this section

Key points

The laws of exponents

  • \(a^m \times a^n = a^{m+n}\), \(a^m \div a^n = a^{m-n}\) and \((a^m)^n = a^{mn}\). The base must be the same for the first two.
  • \((ab)^n = a^n b^n\): the power applies to every factor, including the number. \((2x^3)^3 = 8x^9\), not \(2x^9\).
  • \(a^0 = 1\), and \(a^{-n} = \dfrac{1}{a^n}\). A negative index means a reciprocal. It does not make the number negative.
  • For a fraction, flip it and drop the minus sign: \(\left(\dfrac{a}{b}\right)^{-n} = \left(\dfrac{b}{a}\right)^{n}\).
  • You can only add the powers of terms that are multiplied. \(x^2 + x^3\) does not simplify.

Standard form

  • A number in standard form is \(a\times10^{k}\), with \(1\le a<10\) and k an integer. \(0.000\,047 = 4.7\times10^{-5}\) and \(380\,000 = 3.8\times10^{5}\).
  • To multiply, multiply the a values and add the powers of 10. To divide, divide the a values and subtract the powers.
  • To add or subtract, first write both numbers with the same power of 10, then add the a values.
  • Check that a is between 1 and 10 at the end. \(0.8\times10^{19}\) is not in standard form: it is \(8\times10^{18}\).
  • Give a to 3 significant figures unless the question says otherwise.

On the calculator

Checking an exact value

  • Enter the power or root as a decimal, for example 125 to the power −2/3, and compare with your exact fraction as a decimal.
  • Put the index in brackets: the whole fraction is the power.
  • The calculator shows a decimal. In an exact question, write the fraction or surd, not the decimal.

Screenshot and video to come

Checking an exponential equation

  • Solve for x with the calculator's solver, or graph both sides and find the intersection.
  • Compare the solution with the one you found by writing both sides with one base.

Screenshot and video to come

Standard form on the calculator

  • Enter \(a\times10^{k}\) with the calculator's power-of-ten entry, and put brackets round each number before you multiply or divide.
  • The calculator may show your answer as a decimal or with an E. Write it as \(a\times10^{k}\) in your answer.
  • Compare with your hand calculation: the a values and the powers of 10 should agree.

Screenshot and video to come

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Index laws with a negative power

\(\text{Write } \frac{(2x^3)^3}{4x^{-2}} \text{ in the form } ax^p.\) Find p.

Working
  1. Power of a product

    \((2x^3)^3 = 2^3 \times x^{9} = 8x^9\)

    Cube both the 2 and the x³. Powers of powers multiply: 3 × 3 = 9.

  2. Divide

    \(\frac{8x^9}{4x^{-2}} = 2x^{9-(-2)}\)

    Divide the numbers: 8 ÷ 4 = 2. Subtract the powers: 9 − (−2).

  3. Answer

    \(2x^{11}\)

    So a = 2 and p = 11.

Video walkthrough coming
A negative index on a fraction

\(\text{Find the exact value of } \left(\frac{5}{2}\right)^{-2}.\)

Working
  1. Flip

    \(\left(\frac{5}{2}\right)^{-2} = \left(\frac{2}{5}\right)^{2}\)

    A negative index means the reciprocal. Flip the fraction and make the index positive.

  2. Square

    \(\frac{2^2}{5^2} = \frac{4}{25}\)

    Square the top and the bottom.

Video walkthrough coming
Dividing numbers in standard form

A grain of sand has a mass of \(2.4\times10^{-4}\) g. A water molecule has a mass of \(3.0\times10^{-23}\) g. How many water molecules have the same mass as the grain? Give the answer in the form \(a\times10^{k}\), and write down a.

Working
  1. Divide

    \(\dfrac{2.4\times10^{-4}}{3.0\times10^{-23}}\)

    Divide the grain's mass by the mass of one molecule.

  2. The a values

    \(2.4\div3.0=0.8\)

    Divide the a values.

  3. The powers

    \(10^{-4}\div10^{-23}=10^{-4-(-23)}=10^{19}\)

    Subtract the powers. Take care with two negatives.

  4. Standard form

    \(0.8\times10^{19}=8\times10^{18}\)

    0.8 is less than 1, so move one power of 10. Answer: \(8\times10^{18}\) molecules.

Video walkthrough coming