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Algebraic Roots & Indices (Cambridge (CIE) IGCSE Maths)

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  • Write down the index law for a to the power of m cross times a to the power of n.

    The index law is: a to the power of m cross times a to the power of n equals a to the power of m plus n end exponent.

    You add the powers when multiplying two powers with the same base.

    E.g. a cubed cross times a to the power of 5 equals a to the power of 8.

  • Write down the index law for a to the power of m over a to the power of n.

    The index law is: a to the power of m over a to the power of n equals a to the power of m minus n end exponent.

    You subtract the powers when dividing two powers with the same base.

    E.g. f to the power of 9 over f cubed equals f to the power of 6.

  • Write down the index law for open parentheses a to the power of m close parentheses to the power of n.

    The index law is: open parentheses a to the power of m close parentheses to the power of n equals a to the power of m n end exponent.

    You multiply the powers when raising a to the power of m to the power n.

    E.g. open parentheses g to the power of 4 close parentheses to the power of 8 equals g to the power of 32.

  • Write down the index law for a to the power of 1 over m end exponent.

    The index law is: a to the power of 1 over m end exponent equals m-th root of a.

    A fractional power of 1 over m means the m to the power of th root.

    E.g. x to the power of 1 fifth end exponent equals fifth root of x.

  • Write down the index law for open parentheses a b close parentheses to the power of n.

    The index law is: open parentheses a b close parentheses to the power of n equals a to the power of n b to the power of n.

    The power outside acts on every term inside, when the terms inside are multiplied together.

    E.g. open parentheses t y close parentheses cubed equals t cubed y cubed.

  • Write down the index law for a to the power of negative m end exponent.

    The index law is: a to the power of negative m end exponent equals 1 over a to the power of m.

    A negative power means 1 over the positive power.

    E.g. h to the power of negative 2 end exponent equals 1 over h squared.

  • Write down the index law for a to the power of 1.

    The index law is: a to the power of 1 equals a.

    Anything to the power 1 is itself.

    E.g. w to the power of 1 equals w.

  • Write down the index law for a to the power of 0.

    The index law is: a to the power of 0 equals 1.

    Anything to the power 0 is 1.

    E.g. y to the power of 0 equals 1.

  • True or False?

    a to the power of m over n end exponent equals m-th root of a to the power of n end root

    False.

    The order of m and n are incorrect.

    The correct index law is a to the power of m over n end exponent equals n-th root of a to the power of m end root.

    E.g. x to the power of 3 over 4 end exponent equals fourth root of x cubed end root.

  • True or False?

    a to the power of m over n end exponent can be thought of in two different ways, using powers and roots.

    True.

    You can swap the order of the n to the power of th root and the power of m.

    So  a to the power of m over n end exponent equals n-th root of a to the power of m end root  is the first way to think about it, and  a to the power of m over n end exponent equals open parentheses n-th root of a close parentheses to the power of m is the second way to think about it.

    E.g. y to the power of 3 over 5 end exponent equals fifth root of y cubed end root equals open parentheses fifth root of y close parentheses cubed.

  • When talking about indices, what is the base?

    The base is the number or letter being acted on by the power.

    For example, in a to the power of n, the base is a and the power is n.

  • What is the first step to solving the equation 3 to the power of x equals 27 to the power of x plus 1 end exponent?

    You need both sides to be over the same base.

    So the first step is to change the base of 27 to 3, using the fact 27 = 33.

    (Alternatively, you could change the base of 3 to 27, as 3 equals 27 to the power of 1 third end exponent).

  • True or False?

    open parentheses a over b close parentheses to the power of negative n end exponent equals open parentheses b over a close parentheses to the power of n

    True.

    A fraction to a negative power is the same as the flipped fraction to the positive power, open parentheses a over b close parentheses to the power of negative n end exponent equals open parentheses b over a close parentheses to the power of n.

    This is a great trick to use in exams!

Previous:Algebra ToolkitNext:Expanding & Factorising Brackets

    1. Topic 1

      Number Toolkit

    2. Topic 2

      Set Notation & Venn Diagrams

    3. Topic 3

      Prime Factors, HCF & LCM

    4. Topic 4

      Powers, Roots & Standard Form

    5. Topic 5

      Fractions Toolkit

    6. Topic 6

      Percentages Toolkit

    7. Topic 7

      Simple & Compound Interest, Growth & Decay

    8. Topic 8

      Surds

    9. Topic 9

      Working with FDP

    10. Topic 10

      Working with Ratios

    11. Topic 11

      Compound Measures (Speed, Density, Pressure)

    12. Topic 12

      Time, Currency & Conversions

    13. Topic 13

      Rounding, Estimation & Bounds

    14. Topic 14

      Using a Calculator

    1. Topic 1

      Algebra Toolkit

    2. Topic 2

      Algebraic Roots & Indices

    3. Topic 3

      Expanding & Factorising Brackets

    4. Topic 4

      Linear Equations & Inequalities

    5. Topic 5

      Quadratic Equations

    6. Topic 6

      Rearranging Formulas

    7. Topic 7

      Simultaneous Equations

    8. Topic 8

      Algebraic Fractions

    9. Topic 9

      Forming & Solving Equations

    10. Topic 10

      Functions

    11. Topic 11

      Sequences

    12. Topic 12

      Proportion

    1. Topic 1

      Coordinate Geometry

    2. Topic 2

      Linear Graphs

    3. Topic 3

      Quadratic Graphs

    4. Topic 4

      Further Graphs & Tangents

    5. Topic 5

      Solving & Graphing Inequalities

    6. Topic 6

      Real-Life Graphs

    7. Topic 7

      Differentiation

    1. Topic 1

      Geometry Toolkit

    2. Topic 2

      Angles in Polygons & Parallel Lines

    3. Topic 3

      Bearings, Constructions & Scale Drawings

    4. Topic 4

      Circle Theorems

    1. Topic 1

      Area & Perimeter

    2. Topic 2

      Circles, Arcs & Sectors

    3. Topic 3

      Volume & Surface Area

    4. Topic 4

      Congruence & Similarity

    1. Topic 1

      Right-Angled Triangles (Pythagoras & Trigonometry)

    2. Topic 2

      Sine, Cosine Rule & Area of Triangles

    3. Topic 3

      3D Pythagoras & Trigonometry

    4. Topic 4

      Trigonometric Graphs & Equations

    1. Topic 1

      Vectors

    2. Topic 2

      Transformations

    1. Topic 1

      Basic Probability

    2. Topic 2

      Probability Diagrams (Tree & Venn Diagrams)

    3. Topic 3

      Conditional Probability

    1. Topic 1

      Averages & Range

    2. Topic 2

      Statistical Diagrams

    3. Topic 3

      Histograms

    4. Topic 4

      Cumulative Frequency

    5. Topic 5

      Scatter Graphs & Correlation