Study Companion
Topic 02 | Content: Year 9 extension

Index laws extended

Selective prep: negative and zero indices, combining the index laws in one expression, and simple algebraic powers.

35-50 min Printable practice Answer key Challenge included
How to use this page

Read the explanation, work through the examples, then complete the core practice before printing.

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What you will learn

1. Recap: the three core index laws

Core index laws (Year 8)

Product
am×an=am+n.a^m \times a^n = a^{m+n}.
Quotient
aman=am−n.\dfrac{a^m}{a^n} = a^{m-n}.
Power of a power
(am)n=amn.(a^m)^n = a^{mn}.
Power of a product
(ab)n=anbn.(ab)^n = a^n b^n.

2. Zero index

Any non-zero base to the power 00 equals 11.

Zero index
a0=1(a≠0).a^0 = 1 \quad (a \neq 0).

Why? Using the quotient law, amam=am−m=a0\dfrac{a^m}{a^m} = a^{m - m} = a^0. But any number divided by itself is 11, so a0=1a^0 = 1.

Worked example 1 Zero index

Evaluate   70+(3x)0−4\;7^0 + (3x)^0 - 4.

Each power-of-zero term is 11:

1+1−4=−2.1 + 1 - 4 = -2.

3. Negative index

A negative index means reciprocal.

Negative index
a−n=1an,1a−n=an.a^{-n} = \dfrac{1}{a^n}, \qquad \dfrac{1}{a^{-n}} = a^n.

Why? Using the quotient law, a3a5=a3−5=a−2\dfrac{a^3}{a^5} = a^{3 - 5} = a^{-2}. But a3a5=1a2\dfrac{a^3}{a^5} = \dfrac{1}{a^2}. So a−2=1a2a^{-2} = \dfrac{1}{a^2}.

Worked example 2 Rewrite with positive indices

Rewrite   2−3\;2^{-3} as a fraction.

2−3=123=18.2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}.
Worked example 3 Flipping across the bar

Rewrite   1x−4\;\dfrac{1}{x^{-4}} with a positive index.

A negative index under the bar moves up with a positive sign:

1x−4=x4.\dfrac{1}{x^{-4}} = x^4.

4. Combining the laws

Work in a clear order: clean brackets first, then collect bases, then apply quotient / negative index.

Worked example 4 Mixed operation

Simplify   2a3×3a−5a−2\;\dfrac{2a^3 \times 3a^{-5}}{a^{-2}}, answer with a positive index.

  1. Combine the numerator: 6×a3+(−5)=6a−26 \times a^{3 + (-5)} = 6a^{-2}.
  2. Divide by a−2a^{-2}: 6a−2−(−2)=6a0=66a^{-2 - (-2)} = 6a^0 = 6.
Worked example 5 Power of a power with a negative index

Simplify   (2x−3)2\;(2x^{-3})^2.

Apply the power to each factor: 22×x−3×2=4x−6=4x62^2 \times x^{-3 \times 2} = 4x^{-6} = \dfrac{4}{x^6}.

Worked example 6 Mixed bases

Simplify   3a4b−26a−1b3\;\dfrac{3a^4 b^{-2}}{6a^{-1}b^3} with positive indices.

Handle the numeric part, then each letter:

36=12,a4−(−1)=a5,b−2−3=b−5.\dfrac{3}{6} = \dfrac{1}{2}, \quad a^{4 - (-1)} = a^5, \quad b^{-2 - 3} = b^{-5}.

So the expression is a52b5\dfrac{a^5}{2b^5}.

Practice

Fluency

Evaluate

    1. 505^0.
    2. (−3)0(-3)^0.
    3. 70−207^0 - 2^0.
    4. 2−22^{-2}.
    5. 3−33^{-3}.
    6. 10−1+10−210^{-1} + 10^{-2}.
Fluency

Simplify

    1. x5×x−3x^5 \times x^{-3}.
    2. y2y−4\dfrac{y^2}{y^{-4}}.
    3. (a−2)3(a^{-2})^3.
    4. (2m)−2(2m)^{-2}.
    5. 6a32a7\dfrac{6a^3}{2a^7}.
    6. 4x0+(5x)04x^0 + (5x)^0.
Reasoning

Mixed algebraic

    1. Simplify 8p5q−34p2q−1\dfrac{8p^5 q^{-3}}{4p^2 q^{-1}} with positive indices.
    2. Simplify (3a−1b2)2(3a^{-1} b^2)^2.
    3. Simplify (2x3)24x−1\dfrac{(2x^3)^2}{4x^{-1}}.
    4. Write 1a−5×a−3\dfrac{1}{a^{-5}} \times a^{-3} as a single power of aa.
    5. Show that x−2y3x−4y−1=x2y4\dfrac{x^{-2} y^3}{x^{-4} y^{-1}} = x^2 y^4.

Challenge

Reasoning

Harder reasoning

    1. If ax=8a^x = 8 and ay=2a^y = 2, find ax−ya^{x - y}.
    2. Solve 2n=1322^n = \dfrac{1}{32} for nn.
    3. Simplify (2a−3b2)−2\left(\dfrac{2a^{-3}}{b^2}\right)^{-2} with positive indices.
    4. Given x>0x > 0, order from smallest to largest: x3,x−1,x0,x1/2x^3, x^{-1}, x^0, x^{1/2} when x=4x = 4.
Answers

Answer key

Attempt the practice first. When you're ready to check, expand the answers below.

Show the full answer key

Evaluate

    1. 11.
    2. 11.
    3. 1−1=01 - 1 = 0.
    4. 14\dfrac{1}{4}.
    5. 127\dfrac{1}{27}.
    6. 110+1100=0.11\dfrac{1}{10} + \dfrac{1}{100} = 0.11.

Simplify

    1. x5+(−3)=x2x^{5 + (-3)} = x^2.
    2. y2−(−4)=y6y^{2 - (-4)} = y^6.
    3. a−2×3=a−6=1a6a^{-2 \times 3} = a^{-6} = \dfrac{1}{a^6}.
    4. (2m)−2=1(2m)2=14m2(2m)^{-2} = \dfrac{1}{(2m)^2} = \dfrac{1}{4m^2}.
    5. 3a3−7=3a−4=3a43a^{3-7} = 3a^{-4} = \dfrac{3}{a^4}.
    6. 4(1)+1=54(1) + 1 = 5.

Mixed algebraic

    1. 84⋅p5−2q−3−(−1)=2p3q−2=2p3q2\dfrac{8}{4} \cdot p^{5-2} q^{-3-(-1)} = 2p^3 q^{-2} = \dfrac{2p^3}{q^2}.
    2. 9a−2b4=9b4a29a^{-2}b^4 = \dfrac{9b^4}{a^2}.
    3. 4x64x−1=x6−(−1)=x7\dfrac{4x^6}{4x^{-1}} = x^{6 - (-1)} = x^7.
    4. 1a−5×a−3=a5×a−3=a2\dfrac{1}{a^{-5}} \times a^{-3} = a^5 \times a^{-3} = a^2.
    5. x−2x−4⋅y3y−1=x−2−(−4)y3−(−1)=x2y4\dfrac{x^{-2}}{x^{-4}} \cdot \dfrac{y^3}{y^{-1}} = x^{-2 - (-4)} y^{3 - (-1)} = x^2 y^4. ✓\checkmark

Challenge

    1. ax−y=axay=82=4a^{x - y} = \dfrac{a^x}{a^y} = \dfrac{8}{2} = 4.
    2. 132=2−5\dfrac{1}{32} = 2^{-5}, so n=−5n = -5.
    3. (2a−3b2)−2=(b22a−3)2=b44a−6=a6b44\left(\dfrac{2a^{-3}}{b^2}\right)^{-2} = \left(\dfrac{b^2}{2a^{-3}}\right)^2 = \dfrac{b^4}{4a^{-6}} = \dfrac{a^6 b^4}{4}.
    4. With x=4x = 4: x−1=0.25x^{-1} = 0.25, x0=1x^0 = 1, x1/2=2x^{1/2} = 2, x3=64x^3 = 64. Order: x−1<x0<x1/2<x3x^{-1} < x^0 < x^{1/2} < x^3.

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