Module 3 GCSE

Standard form is based on powers of ten

10✖10=10 squared

1=10 to the power of 0

                                         a✖10 to the power of n

a is a value between1 and 10          n is an integer that can be positive or negative

Negative powers are for numbers less than one. A positive power is used for a number greater or equal to 1.

For example...

0.0006= 6✖10 to the power of -4

a.(3✖10 to the power of 4)✖(6✖10 to the power of 8)

Add the powers and multiply the numbers

=18✖10 to the power of 12

In standard form= 1.2✖10 to the power of 13

b.(8✖10 to the power of 5)/(2✖10 to the power of 3)

Divide the numbers and subtract the powers- Subtracting negative numbers is the same as adding, and adding a negative is the same as subtracting

=4✖10 to the power of 2

 difference/ original ✖ 100 

For example...

If the difference= 10

and the original= 189

= 10/189 ✖ 100

= 5.29%

For example...

Cruella gains a 17% decrease in her pay for the next five year how much pay will she recieve in five years.

 Multiplier= 100-17= 83=0.83

7500✖ 0.83 to the power of 5

= 3559.374075

= £3559.37 

Can use decimals to help work out bigger percentages of numbers

For example...(multipliers)

a.1/2% decrease= 100-0.5=0.995

b. 120% increase= 100+120= 22

c. 5% increase= 100+5= 1.05

d. 34 3/4% increase= 100 + 38.75 = 1.3875

e. 12 1/2& decrease= 100- 12.5 = 0.8875

d.(3✖10 to the power of 8)✖(2✖10 to the power of 7) / (3✖10 to the power of 8)+(2✖10 to the power of 7)

=18750000

=√18750000

=4330

e.(5.98✖10 to the power of 24)✖(7.35✖10 to the power of 22)

=4.3953✖10 to the power of 47

f.(7✖10 to the power of -9) / (7✖10 to the power of -5)

=1✖10 to the power of -4

For example....

Share £72 in the ratio 4:5

1. add together the numbers in the ratio

    4+5=9

2.divide the total by that number

    £72 / 9=£8

3.multiply by each part of the ratio

   4✖8= £32    5✖8= £40

For example...

b. 2√18✖3√2

   =6√36

   =6✖6

    =36

c. 3(√2+4)

   =3√2+12

For example...

a. √3(2-√3)

  =2√3-√9

   = 2√3-√3✖3

    = 2√3-3

b. √8 squared + √10 squared =x squared

    = 8+10=x squared  =18=x squared     =√18=x

     =√9✖2

     =3√2

For example...

a. 2√8✖3√2

    =2√4✖2 ✖3√2

    =2✖2√2✖3√2

    =4√2✖3√2

     =12√4

     = 12✖2

     =24

Simplifying surds- look for factor pairs of the value you are given, but one of these values MUST BE SQUARE to help to simplify the expression (don't use 1)

For example....

a. √24= √6✖4

    =√6✖√4

    =2√6

b. √98= √2

    =√2✖√49

    =7√2

Rational= The exact amount.

Rational numbers- these take any value that can be written as, a fraction or a decimal that either terminates or recurs.

Irrational= No exact amount

Irrational numbers- these are numbers/values that cannot be written in the form above and most commonly involve π or surds (√ in the answer)

LCM = Lowest Common Multiple

HCF= Highest Common Factor

LCM:8 and 10

8 16 24 32 40 48 56

10 20 30 40 50 60             Bold numbers equal lowest common multiple=40

HCF:24 and 30

24: 4 and 6, 12 and 2, 1 and 24 

30: 5 and 6, 3 and 10, 1 and 30 Bold numbers = HCF = 6

Prime Factor Decomposition- use factor trees

For example...

                                                      21

                                           21               100

                                    7         3          10        10

                                                        2      5      2   5

The bold numbers = 7✖3✖2✖2✖5✖5

= 7✖3✖2 to the power of 2 ✖5 to the power of 2

3. ( a to the power of m) to the power of n= a to the power off mn

For example...

(k to the power of 3) to the pwoer of 4= k to the power of 12

Times the powers and do the whole numbers to the power

4. a to the power of 0 = 1

Anything to the power of 0 always equals 1

5. a to the power of -2 = You have to do the RECIPRICAL of a and then square it

For example..

2/1=1/2     a to the power of -2= (1/a) squared

Another example....

7 to the power of negative 2 = (1/7)squared = 1/49

.

1. a to the power of m time a to the power of m= a to the power of m plus n

For example...

z to the power of 3 time z to the power of 4= z to the power of 7

Add the powers and time the whole numbers

2. a to the power of m divided by a to the power of n

For example...

g to the power of 7 divided by g to the power of 2= g to the power of 5

Take away the powers and divide the whole numbers

You are not allowed to leave an answer to a problem with a surd as the denominator. You have to rationalise the surd.

For example....

2/√3 Time the surd and the whole number by the surd which equals 2√3/3

Another example...

√3-5/2√3 which equals 3-5√3/ (2 times 3)

= 3-5√3/6

For example...

Write 0.4 recurring as a fraction

x=0.4 recurring    Times both sides by 10

10x=4.4 recurring

Take them away

    x=0.4 recurring

-10x=4.4 recurring

= 9x=4

Which equals x=4/9

6. a to the power of 1/m = m√a

For example...

9 to the power of 1/2 = √9 which equals 3

7. a to the power of n/m= denominator is the root top then tells you what power to apply

For example...

8 to the power of 2/3 = (3√80 squared = 2 squared =4

Any negatives the recipricol is the positive

6. a to the power of 1/m = m√a

For example...

9 to the power of 1/2 = √9 which equals 3

7. a to the power of n/m= denominator is the root top then tells you what power to apply

For example...

8 to the power of 2/3 = (3√80 squared = 2 squared =4

Any negatives the recipricol is the positive

Round each number (in the equation)  to 1 significant place...

a. 119✖5.4/46 = 100✖5/50 =500/50 =10

b. (5.3✖19.8)/(6.2-1.7)= (5✖20)/(6-2) = 100/4 = 25

For example...

1 2/3 + 2 3/4 = 5/3 + 11/4 = 20/12 + 33/12 = 53/12 = 4 5/12

Another example....

5 1/4 - 1 7/8 = 21/4 - 115/8 = 168/32 - 60/32 = 108/32 =  3 12/32 = 3 6/16

Another example....

3 1/2 ✖ 1 1/4 = 7/2 ✖ 5/4 = 35/8 = 4 3/8

One more...

2 4/5 / 1/10 = 14/5 / 1/10 = 14/5 ✖ 10/1 = 140/5 = 28

When multiplying with a decimal ignore the decimal points and multiplying the numbers as you normally would. Add the decimal back into your answer. You answer will have as many decimal points as the two original numbers combined.

For example......

0.5✖0.5=0.25

Another example....

2.2✖0.02=0.004

For example...

0.3✖0.5=0.15

Another example...

4.2/0.3=1.4 because you turn them either into both whole numbers or decimals so divide both by ten = 0.42/0.3

Another example....

3.45/0.5= 6.9 because times them both by ten = 34.5/5

Dividing with decimals

For example...(without a decimal)

15√645     -60         =45        - 45         =0

This is one of the easiest ways to divide with decimals you just apply this knowledge and add a decimal to the problem

For example...

Write 0.231 and the 31 is recurring as a fraction

x=0.231 with the 31 recurring

Times both sides by 100

100x=23.131 with the 31 recurring        Then take away

    100x=23.131 with the 31 recurring-         x=00.231 with the 31 recurring

Which equals 99x=22.9   Times both sides by ten

990x=229 so as a fraction it equals x=229 over 990