12.6.1 Laws of exponents: negative and fractional indices · AI HL · 12.6: Exponents & Logarithms

12.6.1

Laws of exponents: negative and fractional indices

12.6.1

Laws of exponents: negative and fractional indices

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.

Roots and fractional indices

\(x^{1/n} = \sqrt[n]{x}\) and \(x^{m/n} = \left(\sqrt[n]{x}\right)^m\). The bottom of the fraction is the root, the top is the power.
Take the root first: \(125^{2/3} = 5^2 = 25\) is easier than \(\sqrt[3]{125^2}\).
Write every root as a power before you use the laws: \(\sqrt{x} = x^{1/2}\), and \(\dfrac{1}{\sqrt{x}} = x^{-1/2}\).
In \(ax^p\), only the x has the power. \(\dfrac{3}{\sqrt{x}} = 3x^{-1/2}\).
To split a fraction into the form \(ax^p + bx^q\), divide each term on the top by the bottom.

Solving with indices

To solve \(x^{p/q} = k\), raise both sides to the power \(q/p\). Check by putting the answer back.
If \(a^m = a^n\) then \(m = n\) (for \(a > 0\), \(a \neq 1\)). Write both sides as powers of one base, then equate the powers.
Common bases: \(4 = 2^2\), \(8 = 2^3\), \(9 = 3^2\), \(27 = 3^3\), \(25 = 5^2\), and \(\sqrt{a} = a^{1/2}\).
To write y in terms of x (or x in terms of y), get one power on each side, then raise to the reciprocal power.

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

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

Divide the numbers: 8 ÷ 4 = 2. Subtract the powers: 9 − (−2).
Answer
\(2x^{11}\)

So a = 2 and p = 11.

A negative index on a fraction

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

Working

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

Square the top and the bottom.

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

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

Divide the grain's mass by the mass of one molecule.
The a values
\(2.4\div3.0=0.8\)

Divide the a values.
The powers
\(10^{-4}\div10^{-23}=10^{-4-(-23)}=10^{19}\)

Subtract the powers. Take care with two negatives.
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.

A fractional and negative index

\(\text{Find the exact value of } 125^{-\frac{2}{3}}.\)

Working

Negative index
\(125^{-\frac{2}{3}} = \frac{1}{125^{\frac{2}{3}}}\)

Deal with the minus sign first: take the reciprocal.
Root first
\(125^{\frac{2}{3}} = \left(\sqrt[3]{125}\right)^2 = 5^2 = 25\)

The denominator of the index is the root, the numerator is the power. Taking the root first keeps the numbers small.
Answer
\(\frac{1}{25}\)

The value is exact, so leave it as a fraction.

Roots and powers of x

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

Working

Root as a power
\(\sqrt{x} = x^{\frac{1}{2}}\)

Write every root as a fractional power first.
Divide
\(\frac{3x^2}{x^{1/2}} = 3x^{2-\frac{1}{2}}\)

Subtract the powers. The 3 stays in front.
Answer
\(3x^{\frac{3}{2}}\)

a = 3 and p = 3/2.

A power of y in terms of x

\(\text{Given that } y=\frac{4}{x\sqrt{x}}, \text{ write } y^2 \text{ in the form } ax^k.\) Find k.

Working

Write y with one power
\(y = 4x^{-\frac{3}{2}}\)

x√x = x^1 × x^(1/2) = x^(3/2). It is on the bottom, so the power is negative.
Square both parts
\(y^2 = 4^2 \times \left(x^{-\frac{3}{2}}\right)^2\)

Square the 4 and multiply the power by 2.
Answer
\(16x^{-3}\)

a = 16 and k = −3.

Solving x to a fractional power

\(\text{Solve } x^{\frac{3}{2}} = 64.\)

Working

Reciprocal power
\(\left(x^{\frac{3}{2}}\right)^{\frac{2}{3}} = 64^{\frac{2}{3}}\)

Raise both sides to the power 2/3. This turns x^(3/2) into x.
Evaluate
\(64^{\frac{2}{3}} = \left(\sqrt[3]{64}\right)^2 = 4^2 = 16\)

Root first, then the power.
Answer
\(x = 16\)

Check: 16^(3/2) = 4³ = 64.

Using a common base

\(\text{Solve } \left(\sqrt{5}\right)^x = 25^{x-3}.\)

Working

Write both sides as powers of 5
\(5^{\frac{x}{2}} = \left(5^2\right)^{x-3} = 5^{2(x-3)}\)

√5 = 5^(1/2), so (√5)^x = 5^(x/2). 25 = 5².
Equal powers
\(\frac{x}{2} = 2x - 6\)

Same base on both sides, so the powers are equal.
Solve
\(x = 2(2x-6) \Rightarrow 3x=12 \Rightarrow x=4\)

Multiply by 2, collect the x terms.

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Exponents & Logarithms

Laws of exponents: negative and fractional indices

By the end of this section

Key points

The laws of exponents

Standard form

Roots and fractional indices

Solving with indices

On the calculator

Checking an exact value

Checking an exponential equation

Standard form on the calculator

Worked examples

Index laws with a negative power

Working

A negative index on a fraction

Working

Dividing numbers in standard form

Working

A fractional and negative index

Working

Roots and powers of x

Working

A power of y in terms of x

Working

Solving x to a fractional power

Working

Using a common base

Working

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