Repeated multiplication
\(5\times5\times5\times5=5^4\)
The base is 5 and it appears four times.
Chapter 7
Indices 1
Use index notation efficiently and apply the index laws with positive, zero, negative and fractional powers. The key is to recognise when bases are the same and what each type of index means.
By the end of this chapter, you should be able to
- identify the base and index and write repeated multiplication in index form;
- use the multiplication law for terms with the same base;
- use the division law for terms with the same base;
- simplify a power raised to another power;
- understand and use the zero index;
- rewrite negative indices as reciprocals;
- interpret fractional indices as roots and powers;
- combine index laws in multi-step calculations and use a scientific calculator when appropriate.
1
Subtopic 1
Simplifying numbers with indices
Index notation is a shorthand for repeated multiplication. In the expression \(a^n\), \(a\) is the base and \(n\) is the index (or power).
Index 1
\(7=7^1\)
A number written without a visible index has index 1.
Different bases
\(2^3\times5^2\)
Do not combine powers just because they are next to each other; the bases differ.
Exam method
Count repeated factors of each base; do not combine different bases without a valid rule.
Worked example · Easy
Write in index form
Write \(3\times3\times3\times7\times7\) using indices.
There are three factors of 3 and two factors of 7.
\[\boxed{3^3\times7^2}\]Worked example · Medium
Evaluate a power
Work out \(2^6\).
\[2^6=2\times2\times2\times2\times2\times2=\boxed{64}\]
Worked example · Exam-style extension
Group repeated factors correctly
Express \(2\times2\times5\times2\times5\times5\times5\) in index notation.
There are three factors of 2 and four factors of 5.
\[\boxed{2^3\times5^4}\]
Common mistake
\(3^4\) means four factors of 3, not \(3\times4\). So \(3^4=81\), not 12.
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Write \(4\times4\times4\times4\) using indices.
Show Answer
Four factors of 4 give \(\boxed{4^4}\).
2Apply the method[4 marks]
Write \(3\times7\times3\times7\times7\) in index form.
Show Answer
There are two 3s and three 7s.
\[\boxed{3^2\times7^3}\]
3Exam-style challenge[5 marks]
Simplify \(10\times10\times2\times2\times10\times2\) into a product of powers. Do not expand it.
Show Answer
Group equal bases: three 10s and three 2s.
\[\boxed{10^3\times2^3}\]
2
Subtopic 2
Multiplying numbers in index form
When multiplying powers with the same base, add the indices because you are joining the repeated factors.
Multiplication and power-of-a-power laws
\[\boxed{a^m\times a^n=a^{m+n}}\]\[\boxed{(a^m)^n=a^{mn}}\]These laws apply to the same base. If bases differ, keep them separate unless you can first rewrite them with a common base.
Exam method
Add powers only when multiplying identical bases; multiply powers when raising a power to another power.
Worked example · Easy
Same-base multiplication
Simplify \(5^3\times5^7\).
\[5^3\times5^7=5^{3+7}=\boxed{5^{10}}\]
Worked example · Medium
Include an index of 1
Simplify \(8^4\times8\times8^2\).
Write \(8\) as \(8^1\).
\[8^4\times8^1\times8^2=8^{4+1+2}=\boxed{8^7}\]Worked example · Medium
Power of a power
Simplify \((3^4)^5\).
\[(3^4)^5=3^{4\times5}=\boxed{3^{20}}\]
Worked example · Exam-style
Keep different bases separate
Simplify \(2^5\times3^4\times2^7\times3\).
Combine the powers of 2 and powers of 3 separately.
\[2^{5+7}\times3^{4+1}=\boxed{2^{12}\times3^5}\]Worked example · Exam-style extension
Combine multiple index operations
Simplify \((3^2)^4\times3^5\).
Power of a power: \((3^2)^4=3^{2\times4}=3^8\).
Product of same bases: \(3^8\times3^5=\boxed{3^{13}}\).
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Simplify \(6^3\times6^4\).
Show Answer
Add the powers: \(3+4=7\).
\[\boxed{6^7}\]
2Apply the method[4 marks]
Simplify \((5^3)^2\).
Show Answer
Multiply the indices: \(3\times2=6\).
\[\boxed{5^6}\]
3Exam-style challenge[5 marks]
Give \(2^4\times3^2\times2^5\times3\) as a product of powers.
Show Answer
Group equal bases; \(3=3^1\).
\[\boxed{2^9\times3^3}\]
3
Subtopic 3
Dividing numbers in index form
When dividing powers with the same base, subtract the index in the denominator from the index in the numerator. This represents cancelling common factors.
Division law
\[\boxed{a^m\div a^n=a^{m-n}}\qquad(a\neq0)\]Exam method
For division, subtract the entire exponent of the denominator, not just the first power.
Worked example · Easy
Subtract the indices
Simplify \(7^9\div7^4\).
\[7^{9-4}=\boxed{7^5}\]
Worked example · Medium
Simplify the denominator first
Simplify \(\frac{5^{10}}{5^3\times5^2}\).
Denominator:
\[5^3\times5^2=5^5\]Then divide:
\[\frac{5^{10}}{5^5}=5^{10-5}=\boxed{5^5}\]Worked example · Exam-style
Multiply and divide together
Simplify \(\frac{2^8\times2^5}{2^9}\).
\[2^{8+5-9}=\boxed{2^4}\]
Worked example · Exam-style extension
Cancel a product of powers
Simplify \(\frac{8^{11}}{8^3\times8^5}\).
Combine denominator: \(8^{3+5}=8^8\).
Subtract exponents: \(8^{11-8}=\boxed{8^3}\).
High-grade check
For multiplication, indices add. For division, indices subtract. For a power raised to a power, indices multiply.
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Simplify \(10^8\div10^3\).
Show Answer
Subtract 3 from 8.
\[\boxed{10^5}\]
2Apply the method[4 marks]
Simplify \(\frac{7^9}{7^4\times7^2}\).
Show Answer
Denominator is \(7^6\), so exponent \(9-6=3\).
\[\boxed{7^3}\]
3Exam-style challenge[5 marks]
Simplify \(\frac{3^{12}\times3^2}{3^7}\).
Show Answer
Numerator is \(3^{14}\), divide by \(3^7\).
\[\boxed{3^7}\]
4
Subtopic 4
Negative indices
A negative index represents a reciprocal. It does not make the value negative.
Zero index\[\boxed{a^0=1}\quad(a\neq0)\]
This follows from \(a^m\div a^m=a^{m-m}=a^0=1\).
Negative index\[\boxed{a^{-n}=\frac1{a^n}}\]
A negative power moves the base across the fraction line and makes the index positive.
Exam method
A negative index means reciprocal, not a negative number; a nonzero base to the zero power is 1.
Worked example · Easy
Write as a fraction
Write \(4^{-3}\) as a fraction.
\[4^{-3}=\frac1{4^3}=\boxed{\frac1{64}}\]
Worked example · Medium
Negative power of a fraction
Evaluate \(\left(\frac25\right)^{-2}\).
Invert the fraction and make the index positive.
\[\left(\frac25\right)^{-2}=\left(\frac52\right)^2=\boxed{\frac{25}{4}}\]Worked example · Medium
Use the index law
Evaluate \(3^5\times3^{-3}\).
\[3^{5+(-3)}=3^2=\boxed9\]
Worked example · Exam-style
Zero and negative indices together
Evaluate \(2^0+2^{-2}\).
\[2^0=1,\qquad2^{-2}=\frac14\]\[1+\frac14=\boxed{\frac54}\]
Worked example · Exam-style extension
Understand zero and negative powers
Evaluate \(\left(\frac34\right)^{-2}\).
Invert the fraction, then square it.
\[\left(\frac34\right)^{-2}=\left(\frac43\right)^2=\boxed{\frac{16}{9}}\]
Common mistake
\(5^{-2}\neq-25\). The negative sign belongs to the index, so \(5^{-2}=\frac1{25}\).
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Write \(2^{-4}\) as an ordinary fraction.
Show Answer
\[2^{-4}=\frac1{2^4}=\boxed{\frac1{16}}\]
2Apply the method[4 marks]
Evaluate \(9^0+2^{-2}\).
Show Answer
\(9^0=1\) and \(2^{-2}=\frac14\).
\[1+\frac14=\boxed{\frac54}\]
3Exam-style challenge[5 marks]
Simplify \(\frac{5^3}{5^7}\) without negative indices.
Show Answer
Subtract exponents: \(5^{3-7}=5^{-4}\).
\[\boxed{\frac1{5^4}}\]
5
Subtopic 5
Fractional indices
A fractional index represents a root. The denominator tells you the root; the numerator tells you the power.
Fractional-index laws
\[\boxed{a^{1/n}=\sqrt[n]{a}}\]\[\boxed{a^{m/n}=\left(\sqrt[n]{a}\right)^m=\sqrt[n]{a^m}}\]For numerical work it is usually easier to take the root first, provided the root is simple.
Exam method
For an index \(m/n\), take the nth root and raise to the mth power; a minus sign then gives a reciprocal.
Worked example · Easy
Square-root index
Evaluate \(81^{1/2}\).
\[81^{1/2}=\sqrt{81}=\boxed9\]
Worked example · Easy
Cube-root index
Evaluate \(125^{1/3}\).
\[125^{1/3}=\sqrt[3]{125}=\boxed5\]
Worked example · Medium
Fractional power
Evaluate \(64^{2/3}\).
Take the cube root first:
\[64^{2/3}=(\sqrt[3]{64})^2=4^2=\boxed{16}\]Worked example · Medium
Negative fractional index
Evaluate \(25^{-1/2}\).
\[25^{-1/2}=\frac1{25^{1/2}}=\frac1{\sqrt{25}}=\boxed{\frac15}\]
Worked example · Exam-style
Combine index ideas
Evaluate \(16^{3/2}\times2^{-2}\).
\[16^{3/2}=(\sqrt{16})^3=4^3=64\]\[2^{-2}=\frac14\]\[64\times\frac14=\boxed{16}\]
Worked example · Exam-style extension
Solve fractional-index calculations
Evaluate \(32^{3/5}\).
Denominator 5 means fifth root: \(\sqrt[5]{32}=2\).
Numerator 3 means cube the result.
\[32^{3/5}=2^3=\boxed8\]
Using a calculator
Scientific calculators usually have a power key such as \(x^y\) or \(^\). Enter fractional and negative indices using brackets where needed. For example, enter \(125^{(2/3)}\), not a sequence that could be interpreted as \((125^2)/3\).
Check your understanding
Try each question before opening its worked solution.
1Confidence builder[3 marks]
Evaluate \(64^{1/3}\).
Show Answer
The cube root of 64 is 4.
\[\boxed4\]
2Apply the method[4 marks]
Calculate \(81^{3/4}\).
Show Answer
The fourth root of 81 is 3; cube it.
\[3^3=\boxed{27}\]
3Exam-style challenge[5 marks]
Evaluate \(16^{-3/4}\).
Show Answer
Fourth root of 16 is 2, then cube to get 8.
Negative index means reciprocal.
\[\boxed{\frac18}\]
Chapter 7 practice
Start with index notation and the basic laws, then move to negative and fractional powers. Keep the answers closed until you have attempted each question.
A. Build confidence
1Index notation[3 marks]
Write \(6\times6\times6\times5\times5\) using indices.
Show Answer
There are three factors of 6 and two factors of 5.
\[\boxed{6^3\times5^2}\]2Evaluate powers[3 marks]
Work out a) \(3^4\) b) \(2^7\) c) \(10^3\).
Show Answer
a) \(3^4=81\)
b) \(2^7=128\)
c) \(10^3=1000\)
3Multiply same bases[2 marks]
Simplify \(4^5\times4^3\).
Show Answer
\[4^{5+3}=\boxed{4^8}\]
4Power of a power[2 marks]
Simplify \((7^3)^4\).
Show Answer
\[(7^3)^4=7^{3\times4}=\boxed{7^{12}}\]
B. Apply the laws
5Division[2 marks]
Simplify \(9^{11}\div9^6\).
Show Answer
\[9^{11-6}=\boxed{9^5}\]
6Mixed multiply/divide[3 marks]
Simplify \(\frac{3^7\times3^5}{3^9}\).
Show Answer
\[3^{7+5-9}=\boxed{3^3}\]
7Different bases[3 marks]
Simplify \(2^4\times5^3\times2^6\times5^2\).
Show Answer
\[2^{4+6}\times5^{3+2}=\boxed{2^{10}\times5^5}\]
8Zero index[2 marks]
Work out \(13^0+4^0\).
Show Answer
\[13^0=1,\quad4^0=1\]\[1+1=\boxed2\]
C. Negative indices
9Write as a fraction[2 marks]
Write \(7^{-2}\) as a fraction.
Show Answer
\[7^{-2}=\frac1{7^2}=\boxed{\frac1{49}}\]
10Fraction with negative index[3 marks]
Evaluate \(\left(\frac34\right)^{-2}\).
Show Answer
\[\left(\frac34\right)^{-2}=\left(\frac43\right)^2=\boxed{\frac{16}{9}}\]
11Combine powers[3 marks]
Evaluate \(5^6\times5^{-4}\).
Show Answer
\[5^{6-4}=5^2=\boxed{25}\]
D. Fractional indices
12Roots as indices[3 marks]
Evaluate a) \(49^{1/2}\) b) \(216^{1/3}\).
Show Answer
a) \(\sqrt{49}=\boxed7\)
b) \(\sqrt[3]{216}=\boxed6\)
13Fractional power[3 marks]
Evaluate \(27^{2/3}\).
Show Answer
\[27^{2/3}=(\sqrt[3]{27})^2=3^2=\boxed9\]
14Negative fractional power[4 marks]
Evaluate \(64^{-2/3}\).
Show Answer
\[64^{2/3}=(\sqrt[3]{64})^2=4^2=16\]\[64^{-2/3}=\boxed{\frac1{16}}\]
E. Exam-style challenge
15Multi-step indices[4 marks]
Simplify \(\frac{(2^3)^4\times2^{-5}}{2^2}\), giving the answer as a single power of 2.
Show Answer
\[(2^3)^4=2^{12}\]\[\frac{2^{12}\times2^{-5}}{2^2}=2^{12-5-2}=\boxed{2^5}\]
16Evaluate exactly[5 marks]
Work out \(81^{3/4}\times3^{-2}\).
Show Answer
\[81^{3/4}=(\sqrt[4]{81})^3=3^3=27\]\[3^{-2}=\frac19\]\[27\times\frac19=\boxed3\]
17Recognise an equivalent form[4 marks]
Write \(\sqrt[5]{x^3}\) using a fractional index.
Show Answer
The fifth root gives denominator 5 and the power 3 gives numerator 3.
\[\boxed{x^{3/5}}\]18Explain the error[4 marks]
A student says \(4^3+4^2=4^5\). Explain why this is wrong and find the correct value.
Show Answer
The law for adding indices applies to multiplication, not addition.
\[4^3+4^2=64+16=\boxed{80}\]Chapter 7 high-grade checklist
Notation
I can identify the base and index and translate repeated multiplication into index notation.
Multiplication
For the same base, I add indices; for a power of a power, I multiply indices.
Division
For the same base, I subtract the denominator index from the numerator index.
Zero and negative
I know \(a^0=1\) and \(a^{-n}=1/a^n\).
Fractional indices
I use the denominator as the root and the numerator as the power.
Exam discipline
I check that the bases match before applying an index law and keep exact answers where appropriate.
Illustrations are taken only from actual Chapter 7 diagrams where present; explanations and practice questions are newly written. Text-only book pages are not reproduced as images.