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GCSEMathsAQAHigherFlashcards1. NumberPowers, Roots & Standard Form

Powers, Roots & Standard Form (AQA GCSE Maths)

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  • FrontPowers & Roots

    What is a square root of a number?

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  • What is a square root of a number?

    A square root of a number is a value that, when squared, gives the original number.

    E.g. 3 is the square root of 9 because 32 = 3 x 3 = 9.

  • True or False?

    Every positive number has two square roots.

    True.

    Every positive number has two square roots, one positive and one negative.

    E.g. square root of 64 equals 8 and negative 8

  • True or False?

    Cube roots of negative numbers do not exist.

    False.

    The cube root of a negative number will also be a negative number.

    For example, -5 is a cube root of -125.

  • Define the term reciprocal.

    The reciprocal of a number is the number that you multiply it by to get 1.

    For example, 2 over 3 is the reciprocal of 3 over 2.

  • How do I estimate a root, e.g. the square root of 27?

    You can estimate the root of a number by finding the closest integer roots either side of it.

    E.g. To find square root of 27

    square root of 25 equals 5 and square root of 36 equals 6

    So square root of 27 is between 5 and 6

  • What number do you get when you raise any non-zero number to the power of zero, e.g. 20?

    Any non-zero number raised to the power of 0 is equal to 1.

    E.g. 20 = 1.

  • True or False?

    If you raise a non-zero number to the power of 1, you get 1.

    False.

    Any number raised to the power 1 is just itself.

    E.g. 6 to the power of 1 equals 6.

  • What do you get if you raise a non-zero number to the power of -1,
    e.g. 3-1 ?

    If you raise a non-zero number to the power of -1 you get the reciprocal of the number.

    E.g. 3 to the power of negative 1 end exponent equals 1 third.

  • What do you get if you raise a positive number to the power of ½,
    e.g. 51/2 ?

    If you raise a non-zero number to the power of ½ you get its positive square root.

    E.g. 5 to the power of 1 half end exponent equals square root of 5.

  • What is the index law for a to the power of m cross times a to the power of n?

    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

    If you multiply two powers with the same base number, you add the indices together.

  • What is the index law for a to the power of m divided by a to the power of n?

    a to the power of m divided by a to the power of n equals a to the power of m minus n end exponent

    If you divide two powers with the same base number, you subtract one index from the other.

  • What is the index law for open parentheses a to the power of m close parentheses to the power of n?

    open parentheses a to the power of m close parentheses to the power of n equals a to the power of m cross times n end exponent

    If you raise a power to another power, you multiply the indices.

  • True or False?

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

    False.

    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

    The denominator of an index is the root and the numerator of an index is the power.

  • What does a number written in standard form look like?

    A number in standard form looks like a cross times 10 to the power of n, e.g. 5.2 cross times 10 cubed.

    It is a number between 1 and 10 multiplied by a power of ten.

  • When a number is written in standard form a cross times 10 to the power of n, which values can a be?

    For a cross times 10 to the power of n, a is at least 1 and less than 10, 1 less or equal than a less than 10.

    This means there is one non-zero digit before the decimal point.

  • True or False?

    When a number is written in standard form a cross times 10 to the power of n, n can be a fraction.

    False.

    For a number written in standard form, a cross times 10 to the power of n, the value n must be an integer.

    It can be zero, positive or negative.

  • When a value between 0 and 1, is written in standard form a cross times 10 to the power of n, what type of number will n be?

    If a value is between 0 and 1 then when it is written in standard form, n will be a negative integer.

    E.g. 0.083 would be written as 8.3 x 10-2.

  • How do you multiply two numbers that are written in standard form and give the answer in standard form without a calculator?

    E.g. open parentheses 2 cross times 10 cubed close parentheses cross times open parentheses 6 cross times 10 to the power of 5 close parentheses

    Multiply together the values of a from both numbers and write this in standard form if needed, e.g. 2 cross times 6 equals 12 equals 1.2 cross times 10 to the power of 1.

    Add together the indices for the powers of 10 from each number, e.g. 10 cubed cross times 10 to the power of 5 equals 10 to the power of 8.

    Multiply the two parts together, e.g. 1.2 cross times 10 to the power of 1 cross times 10 to the power of 8.

    Simplify if necessary, e.g. 1.2 cross times 10 to the power of 9.

  • How do you divide two numbers that are written in standard form and give the answer in standard form without a calculator?

    E.g. open parentheses 2 cross times 10 to the power of 9 close parentheses divided by open parentheses 8 cross times 10 squared close parentheses

    Divide the values of a for both numbers and write this in standard form if needed, e.g. 2 divided by 8 equals 0.25 equals 2.5 cross times 10 to the power of negative 1 end exponent

    Subtract the indices for the powers of 10 for both numbers, e.g. 10 to the power of 9 divided by 10 squared equals 10 to the power of 7

    Multiply the two parts together, e.g. 2.5 cross times 10 to the power of negative 1 end exponent cross times 10 to the power of 7

    Simplify if necessary, e.g. 2.5 cross times 10 to the power of 6

  • How do you add two numbers that are written in standard form and give the answer in standard form without a calculator?

    E.g. open parentheses 4 cross times 10 to the power of 7 close parentheses plus open parentheses 9 cross times 10 cubed close parentheses

    Rewrite the smaller number so that it has the same power as the larger number, e.g. open parentheses 4 cross times 10 to the power of 7 close parentheses plus open parentheses 0.000 space 9 cross times 10 to the power of 7 close parentheses

    Add the values of a and multiply by the same power of 10, e.g. open parentheses 4 cross times 10 to the power of 7 close parentheses plus open parentheses 0.000 space 9 cross times 10 to the power of 7 close parentheses equals 4.000 space 9 cross times 10 to the power of 7

  • How do you subtract two numbers that are written in standard form and give the answer in standard form without a calculator?

    E.g. open parentheses 7 cross times 10 to the power of 4 close parentheses minus open parentheses 6 cross times 10 squared close parentheses

    Rewrite the smaller number so that it has the same power as the larger number, e.g. open parentheses 7 cross times 10 to the power of 4 close parentheses minus open parentheses 0.06 cross times 10 to the power of 4 close parentheses

    Subtract the values of a and multiply by the same power of 10, e.g. open parentheses 7 cross times 10 to the power of 4 close parentheses minus open parentheses 0.06 cross times 10 to the power of 4 close parentheses equals 6.94 cross times 10 to the power of 4

Previous:Prime Factors, HCF & LCMNext:Fractions

    1. Topic 1

      Number Operations

    2. Topic 2

      Prime Factors, HCF & LCM

    3. Topic 3

      Powers, Roots & Standard Form

    4. Topic 4

      Fractions

    5. Topic 5

      Percentages

    6. Topic 6

      Simple & Compound Interest, Growth & Decay

    7. Topic 7

      Fractions, Decimals & Percentages

    8. Topic 8

      Rounding, Estimation & Bounds

    9. Topic 9

      Surds

    10. Topic 10

      Using a Calculator

    1. Topic 1

      Introduction to Algebra

    2. Topic 2

      Algebraic Roots & Indices

    3. Topic 3

      Expanding Brackets

    4. Topic 4

      Factorising

    5. Topic 5

      Completing the Square

    6. Topic 6

      Algebraic Fractions

    7. Topic 7

      Rearranging Formulas

    8. Topic 8

      Algebraic Proof

    9. Topic 9

      Linear Equations

    10. Topic 10

      Quadratic Equations

    11. Topic 11

      Simultaneous Equations

    12. Topic 12

      Iteration

    13. Topic 13

      Forming & Solving Equations

    14. Topic 14

      Functions

    15. Topic 15

      Coordinate Geometry

    16. Topic 16

      Linear Graphs

    17. Topic 17

      Graphs of Functions

    18. Topic 18

      Equation of a Circle

    19. Topic 19

      Estimating Gradients & Areas under Graphs

    20. Topic 20

      Real-Life Graphs

    21. Topic 21

      Solving Inequalities

    22. Topic 22

      Graphing Inequalities

    23. Topic 23

      Transformations of Graphs

    24. Topic 24

      Sequences

    1. Topic 1

      Ratios

    2. Topic 2

      Problem Solving with Ratios

    3. Topic 3

      Direct & Inverse Proportion

    4. Topic 4

      Standard & Compound Units

    5. Topic 5

      Exchange Rates & Best Buys

    1. Topic 1

      Symmetry & Shapes

    2. Topic 2

      Angles in Polygons & Parallel Lines

    3. Topic 3

      Bearings, Scale Drawing, Constructions & Loci

    4. Topic 4

      Circle Theorems

    5. Topic 5

      Area & Perimeter

    6. Topic 6

      Circles, Arcs & Sectors

    7. Topic 7

      Volume & Surface Area

    8. Topic 8

      Congruence, Similarity & Geometrical Proof

    9. Topic 9

      Area & Volume of Similar Shapes

    10. Topic 10

      Pythagoras & Trigonometry

    11. Topic 11

      Sine, Cosine Rule & Area of Triangles

    12. Topic 12

      3D Pythagoras & Trigonometry

    13. Topic 13

      Vectors

    14. Topic 14

      Transformations

    1. Topic 1

      Introduction to Probability

    2. Topic 2

      Simple Probability Diagrams

    3. Topic 3

      Tree Diagrams

    4. Topic 4

      Combined & Conditional Probability

    1. Topic 1

      Averages, Ranges & Data

    2. Topic 2

      Statistical Diagrams

    3. Topic 3

      Histograms

    4. Topic 4

      Cumulative Frequency & Box Plots

    5. Topic 5

      Scatter Graphs & Correlation