Multiply same base
\(a^m\times a^n=a^{m+n}\)
Add the indices.
Chapter 22
Indices 2
Apply the index laws confidently to algebraic and numerical expressions, including zero, negative and fractional indices, and learn how to rewrite different bases before simplifying.
By the end of this chapter, you should be able to
- multiply and divide algebraic expressions written in index form;
- raise powers and coefficients to another power correctly;
- use the zero-index rule and simplify expressions containing zero powers;
- use negative and fractional indices accurately;
- rewrite numbers with a common base before applying index laws;
- solve simple exponential equations by expressing both sides with the same base.
1
Subtopic 1
Simplifying algebraic expressions using indices
Exam-ready method
Check the base before applying an index rule. Apply powers to every factor inside brackets; unlike powers cannot simply be added.
The same index laws used with numbers also apply to algebraic expressions. The key is to work with the coefficient and each base separately.
Divide same base
\(a^m\div a^n=a^{m-n}\)
Subtract the indices.
Power of a power
\((a^n)^m=a^{nm}\)
Multiply the indices.
Zero index
\(a^0=1\)
For \(a\neq0\).
\(a^3\times a^5\)same base → add indices\(a^{3+5}=a^8\)
\(a^7\div a^2\)same base → subtract indices\(a^{7-2}=a^5\)
\((a^4)^3\)power of a power → multiply indices\(a^{12}\)
Worked example · Easy
Multiply coefficients and powers separately
Simplify \(3a^2\times4a^3\).
Multiply the coefficients:
\[3\times4=12\]Add the indices because the base is the same:
\[a^2\times a^3=a^5\]Answer: \(\boxed{12a^5}\).
Worked example · Easy
Division
Simplify \(\frac{18a^7}{6a^3}\).
\[18\div6=3\]\[a^7\div a^3=a^{7-3}=a^4\]
So the answer is \(\boxed{3a^4}\).
Worked example · Medium
Power of a product
Simplify \((2x^3)^4\).
Raise the coefficient and the power to the fourth power:
\[(2x^3)^4=2^4x^{12}=\boxed{16x^{12}}\]Worked example · Medium
Several letters
Simplify \(\frac{12a^3b^2\times3a^2b^4}{2a^3b^2}\).
First simplify the numerator:
\[12a^3b^2\times3a^2b^4=36a^5b^6\]Now divide:
\[\frac{36a^5b^6}{2a^3b^2}=18a^{5-3}b^{6-2}=\boxed{18a^2b^4}\]Worked example · Medium
Zero index appears
Simplify \(\frac{15a^4b^3}{5a^4b}\).
\[\frac{15a^4b^3}{5a^4b}=3a^{4-4}b^{3-1}=3a^0b^2\]
Since \(a^0=1\):
\[\boxed{3b^2}\]Do not apply index laws to addition
\(4a^2+3a^3\)
These are not like terms, so the expression cannot be simplified by adding the indices.
Worked example · Exam-style
Mixed operations
Simplify \(\frac{(3a^2b)^2\times2a}{6ab^2}\).
First expand the power:
\[(3a^2b)^2=9a^4b^2\]Multiply the numerator:
\[9a^4b^2\times2a=18a^5b^2\]Divide:
\[\frac{18a^5b^2}{6ab^2}=3a^4\]Answer: \(\boxed{3a^4}\).
New worked example · Exam-style
Several rules together
Simplify \(\frac{(3a^2b)^2\,(2ab^3)}{6a^3b^2}\).
Worked method — show your mathematical steps and check restrictions or units.
\[(3a^2b)^2=9a^4b^2\]
\[\frac{9a^4b^2\cdot2ab^3}{6a^3b^2}=3a^{5-3}b^{5-2}\]
\[\boxed{3a^2b^3}\]
Try it yourself — from confidence to examination level
Attempt each question before opening its step-by-step solution.
1Multiply like bases · Build confidence[2 marks]
Simplify \(5p^3\times2p^4\).
Show Answer
Step-by-step solution
\[5\times2=10,\quad p^{3+4}=p^7\]
\[\boxed{10p^7}\]
2Power of a product · Develop skill[3 marks]
Simplify \((4x^2y)^2\).
Show Answer
Step-by-step solution
\[4^2=16,\quad (x^2)^2=x^4,\quad y^2=y^2\]
\[\boxed{16x^4y^2}\]
3Quotient of monomials · Exam-style[4 marks]
Simplify \(\frac{30a^7b^4}{5a^2b}\).
Show Answer
Step-by-step solution
\[\frac{30}{5}=6,\quad a^{7-2}=a^5,\quad b^{4-1}=b^3\]
\[\boxed{6a^5b^3}\]
4Recognise unlike terms · Exam challenge[5 marks]
Can \(3x^2+4x^3\) be combined into \(7x^5\)? Explain.
Show Answer
Step-by-step solution
\[\text{The terms have different powers of }x.\]
\[\boxed{3x^2+4x^3\text{ cannot be combined as like terms.}}\]
2
Subtopic 2
Using the laws of indices with numerical and algebraic expressions
Exam-ready method
Rewrite numbers as powers of the same base before combining; simplify zero, negative and fractional powers carefully, with nonzero-base restrictions where relevant.
Harder index questions combine several laws at once. It is often necessary to rewrite numbers with a common base before simplifying.
Negative index
\(a^{-n}=\frac1{a^n}\)
A negative index means reciprocal, not a negative value.
Fractional index
\(a^{1/n}=\sqrt[n]{a}\)
The denominator of the index indicates the root.
Combined fractional index
\(a^{m/n}=(\sqrt[n]{a})^m\)
Root and power can be applied in either order.
\(8^{3/4}\)rewrite 8 as \(2^3\)\((2^3)^{3/4}=2^{9/4}\)
\(2^3\times4^{3/2}\)rewrite \(4=2^2\)\(2^3\times(2^2)^{3/2}=2^6\)
\(9^n\times27^{2n}\)rewrite with base 3\(3^{2n}\times3^{6n}=3^{8n}\)
Worked example · Easy
Negative index
Write \(x^{-4}\) with a positive index.
\[x^{-4}=\boxed{\frac1{x^4}}\]
Worked example · Medium
Fractional index
Evaluate \(81^{3/4}\).
Take the fourth root first:
\[81^{1/4}=3\]Then cube:
\[3^3=\boxed{27}\]Worked example · Medium
Write as a single power
Simplify \(2^5\times4^{3/2}\).
Rewrite \(4\) as \(2^2\):
\[4^{3/2}=(2^2)^{3/2}=2^3\]Then:
\[2^5\times2^3=\boxed{2^8}\]Worked example · Medium
Algebraic powers with negative indices
Simplify \(4y^2\div2y^{-3}\).
\[4\div2=2\]\[y^2\div y^{-3}=y^{2-(-3)}=y^5\]
Answer: \(\boxed{2y^5}\).
Worked example · Exam-style
Different numerical bases
Write \(9^n\times27^{2n}\) as a single power of 3.
\[9^n=(3^2)^n=3^{2n}\]\[27^{2n}=(3^3)^{2n}=3^{6n}\]
Multiply same base:
\[3^{2n}\times3^{6n}=\boxed{3^{8n}}\]\(2^x=32\)write 32 as a power of 2\(2^x=2^5\Rightarrow x=5\)
\(16^x=\frac14\)use base 2\(2^{4x}=2^{-2}\Rightarrow x=-\frac12\)
Worked example · Exam-style
Solve an exponential equation
Solve \(64^x=4\).
Write both sides with base 2:
\[64=2^6,\qquad4=2^2\]\[(2^6)^x=2^2\]\[2^{6x}=2^2\]Equate the indices:
\[6x=2\Rightarrow\boxed{x=\frac13}\]Worked example · Exam-style
Several laws together
Simplify \(\frac{6a^{-3}\times4a^2}{3a^{-2}}\).
Coefficients:
\[\frac{6\times4}{3}=8\]Powers of \(a\):
\[a^{-3}a^2\div a^{-2}=a^{-3+2-(-2)}=a^1\]So the expression simplifies to \(\boxed{8a}\).
High-grade habit
When different bases appear, first ask whether each number can be written as a power of the same base. Once the bases match, the index laws become much easier to apply.
New worked example · Exam-style
Combine fractional and negative indices
Write \(\frac{9^{1/2}\times3^{-2}}{3^{-1}}\) as a single power of 3.
Worked method — show your mathematical steps and check restrictions or units.
\[9^{1/2}=(3^2)^{1/2}=3^1\]
\[\frac{3^{1-2}}{3^{-1}}=3^{-1-(-1)}=3^0\]
\[\boxed{1}\]
Try it yourself — from confidence to examination level
Attempt each question before opening its step-by-step solution.
1Negative index · Build confidence[2 marks]
Write \(2^{-4}\) as a fraction.
Show Answer
Step-by-step solution
\[2^{-4}=\frac1{2^4}\]
\[\boxed{\frac1{16}}\]
2Fractional index · Develop skill[3 marks]
Evaluate \(81^{3/4}\).
Show Answer
Step-by-step solution
\[81^{1/4}=3\]
\[81^{3/4}=3^3=\boxed{27}\]
3Same numerical base · Exam-style[4 marks]
Simplify \(8^2\times4^{-1}\) as a power of 2.
Show Answer
Step-by-step solution
\[8^2=(2^3)^2=2^6,\quad4^{-1}=(2^2)^{-1}=2^{-2}\]
\[\boxed{2^4=16}\]
4Algebraic negative power · Exam challenge[5 marks]
Simplify \(\frac{x^{-2}y^3}{x^2y^{-1}}\) using positive indices.
Show Answer
Step-by-step solution
\[x^{-2-2}y^{3-(-1)}=x^{-4}y^4\]
\[\boxed{\frac{y^4}{x^4}},\quad x\ne0\]
Chapter 22 practice
Begin with the basic algebraic laws, then move to negative and fractional indices, common-base rewriting and simple exponential equations. Keep answers closed until you have tried each question.
A. Build confidence
1Multiply same base[2 marks]
Simplify \(5a^3\times2a^4\).
Show Answer
Method: Rewrite matching bases where useful, apply the relevant index law to each factor, and express the final answer in its simplest form.
\[5\times2=10\]\[a^3a^4=a^7\]\(\boxed{10a^7}\)
2Divide same base[2 marks]
Simplify \(\frac{21x^8}{7x^3}\).
Show Answer
Method: Rewrite matching bases where useful, apply the relevant index law to each factor, and express the final answer in its simplest form.
\[21\div7=3,\qquad x^{8-3}=x^5\]\(\boxed{3x^5}\)
3Power of a power[2 marks]
Simplify \((4y^2)^3\).
Show Answer
Method: Rewrite matching bases where useful, apply the relevant index law to each factor, and express the final answer in its simplest form.
\[4^3=64,\qquad(y^2)^3=y^6\]\(\boxed{64y^6}\)
4Zero index[2 marks]
Simplify \(7p^0q^3\).
Show Answer
Method: Rewrite matching bases where useful, apply the relevant index law to each factor, and express the final answer in its simplest form.
Since \(p^0=1\):
\[7p^0q^3=\boxed{7q^3}\]B. Negative and fractional indices
5Negative index[2 marks]
Write \(m^{-5}\) using only positive indices.
Show Answer
Method: Rewrite matching bases where useful, apply the relevant index law to each factor, and express the final answer in its simplest form.
\[\boxed{\frac1{m^5}}\]6Fractional index[2 marks]
Evaluate \(64^{1/3}\).
Show Answer
Method: Rewrite matching bases where useful, apply the relevant index law to each factor, and express the final answer in its simplest form.
The cube root of 64 is 4.
\[\boxed{4}\]7Fractional power[3 marks]
Evaluate \(32^{2/5}\).
Show Answer
Method: Rewrite matching bases where useful, apply the relevant index law to each factor, and express the final answer in its simplest form.
The fifth root of 32 is 2.
\[32^{2/5}=2^2=\boxed{4}\]8Subtract a negative index[3 marks]
Simplify \(6x^4\div3x^{-2}\).
Show Answer
Method: Rewrite matching bases where useful, apply the relevant index law to each factor, and express the final answer in its simplest form.
\[6\div3=2\]\[x^{4-(-2)}=x^6\]\(\boxed{2x^6}\)
C. Common-base reasoning
9Rewrite a base[3 marks]
Write \(2^4\times8^2\) as a single power of 2.
Show Answer
Method: Rewrite matching bases where useful, apply the relevant index law to each factor, and express the final answer in its simplest form.
\[8^2=(2^3)^2=2^6\]\[2^4\times2^6=\boxed{2^{10}}\]10Variable exponent[3 marks]
Write \(4^n\times8^{2n}\) as a single power of 2.
Show Answer
Method: Rewrite matching bases where useful, apply the relevant index law to each factor, and express the final answer in its simplest form.
\[4^n=(2^2)^n=2^{2n}\]\[8^{2n}=(2^3)^{2n}=2^{6n}\]\[2^{2n}2^{6n}=\boxed{2^{8n}}\]11Cannot be combined[2 marks]
Can \(3^4+3^2\) be written as \(3^6\)? Explain.
Show Answer
No. The rule \(a^m\times a^n=a^{m+n}\) applies to multiplication, not addition.
Indeed, \(3^4+3^2=81+9=90\), while \(3^6=729\).
12Mixed base and fractional index[4 marks]
Simplify \(2^3\times4^{5/2}\) as a single power of 2.
Show Answer
Method: Rewrite matching bases where useful, apply the relevant index law to each factor, and express the final answer in its simplest form.
\[4^{5/2}=(2^2)^{5/2}=2^5\]\[2^3\times2^5=\boxed{2^8}\]D. Exam-style challenge
13Solve for the index[3 marks]
Solve \(3^x=81\).
Show Answer
Method: Rewrite matching bases where useful, apply the relevant index law to each factor, and express the final answer in its simplest form.
\[81=3^4\]So \(3^x=3^4\), hence \(\boxed{x=4}\).
14Fractional solution[4 marks]
Solve \(16^x=8\).
Show Answer
Method: Rewrite matching bases where useful, apply the relevant index law to each factor, and express the final answer in its simplest form.
\[16=2^4,\qquad8=2^3\]\[2^{4x}=2^3\]\[4x=3\Rightarrow\boxed{x=\frac34}\]15Negative solution[4 marks]
Solve \(9^x=\frac1{27}\).
Show Answer
Method: Rewrite matching bases where useful, apply the relevant index law to each factor, and express the final answer in its simplest form.
\[9^x=(3^2)^x=3^{2x}\]\[\frac1{27}=3^{-3}\]\[2x=-3\Rightarrow\boxed{x=-\frac32}\]16Several laws together[5 marks]
Simplify \(\frac{12a^{-2}b^5\times3a^4b^{-1}}{6ab^2}\).
Show Answer
Coefficients:
\[\frac{12\times3}{6}=6\]Powers of \(a\):
\[a^{-2+4-1}=a^1\]Powers of \(b\):
\[b^{5-1-2}=b^2\]So the simplified expression is \(\boxed{6ab^2}\).
Chapter 22 high-grade checklist
Same base first
I only add or subtract indices when the bases are the same and the operation is multiplication or division.
Coefficients separately
I simplify the numerical coefficient separately from the indexed letters.
Power of a power
I multiply indices and also raise any numerical coefficient to the outside power.
Zero and negative indices
I know \(a^0=1\) and \(a^{-n}=1/a^n\).
Fractional indices
I interpret the denominator as a root and the numerator as a power.
Exponential equations
I rewrite both sides with a common base, then equate the indices.