QUICK MATH REVIEW & TIPSQUICK MATH REVIEW & TIPS

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Step into Algebra and Step into Algebra and Conquer itConquer it

ExponentsExponents

Exponents are used to simplify expressions where the same number Exponents are used to simplify expressions where the same number multiplies itself several times.multiplies itself several times.

For example the number 8 can be written as the product 2For example the number 8 can be written as the product 2xx22xx2 or 2·2·22 or 2·2·2

Instead of writing 8=2·2·2 we can shorten the expression by using Instead of writing 8=2·2·2 we can shorten the expression by using exponents. exponents.

So we write 8=So we write 8=2233 {2 {23 3 is short for 2·2·2)is short for 2·2·2)

22 is called the is called the basebase and and 33 is the is the powerpower or or exponentexponent

2233 is read as “is read as “TwoTwo to the power to the power threethree” or “” or “TwoTwo raised to the power raised to the power threethree””

Other examples:Other examples:

27=3·3·3 = 327=3·3·3 = 333

16=2·2·2·2 = 216=2·2·2·2 = 24 4

As you can see the exponent simply shows how many times the base As you can see the exponent simply shows how many times the base multiplies itself to arrive at the product.multiplies itself to arrive at the product.

If you raise any number to the power If you raise any number to the power of 0 you will get of 0 you will get 11

2200= 1= 1

3300= 1 = 1

10010000= 1= 1

xx00 =1 =1

100010000 = 0 = 11

If you raise any number to the powerIf you raise any number to the power

of of 11 you will get the original number you will get the original number

2211 = 2 = 2

3311 = 3 = 3

5511 = 5 = 5

1001001 1 = 100= 100

If you raise any number to the power ofIf you raise any number to the power of

NEGATIVE ONE (NEGATIVE ONE (--1) you will get the1) you will get the

reciprocalreciprocal of the original number of the original number

22-1-1 = = 11

22

33-1-1 = = 1 1

33

If you raise a number to a If you raise a number to a negative power negative power

you get the reciprocal of the original numberyou get the reciprocal of the original number

raised to the raised to the positivepositive of the original power: of the original power:

22--33 == 1 1 == 11

2233 88

33--44 == 1 1 == 1 1

3344 8181

When you multiply exponents that When you multiply exponents that have a common or similar base, their have a common or similar base, their powers simply add up:powers simply add up:

8 =2.2.2 = 28 =2.2.2 = 23 3

27 =3.3.3 =327 =3.3.3 =333

36 = 4.9 =2.2.3.3=236 = 4.9 =2.2.3.3=222.3.322

2233.2.255=2=28 8 =2=2((33++55))

2266.2.2-4-4=2=22 2 =2=2((66-4-4))

Practice QuestionsPractice Questions Write the product 3·3·3·7·7·7 using exponents.

Write 108 as a product of its prime factors in expanded form and then in exponential form.

Find the value of n in 81=3(n-2)

Find the value of d in the below equation if n=7 d=2(n+3).3 (n-4)

If 72=2(x-2)·32, what is the value of x?

You can use You can use any letterany letter to represent an UNKNOWN in a to represent an UNKNOWN in aany Math problem.any Math problem.

For example if we are told that there are 36 students in a class out of which 23 are For example if we are told that there are 36 students in a class out of which 23 are boys and we are required to find the number of girls, we can start out by choosing a boys and we are required to find the number of girls, we can start out by choosing a letter to represent the unknown (in this case the number of girls). Then write down a letter to represent the unknown (in this case the number of girls). Then write down a simple equation for the total number of students in the class with the information we simple equation for the total number of students in the class with the information we know so far.know so far.

If we choose the letter g to represent the number of girls, then we can write:If we choose the letter g to represent the number of girls, then we can write:

23 + g = 3623 + g = 36

where 23 is the number of boyswhere 23 is the number of boys

g is the letter we have selected to represent the number of girlsg is the letter we have selected to represent the number of girls

36 is the total number of students in the class36 is the total number of students in the class

Using letters to represent unknowns comes very handy when dealingUsing letters to represent unknowns comes very handy when dealingwith very long statement math problems. Just follow the statements in the questionwith very long statement math problems. Just follow the statements in the questionpatiently and use letters to represent the unknowns as necessary. You willpatiently and use letters to represent the unknowns as necessary. You willthen be able to write down a mathematical statement in place of the longthen be able to write down a mathematical statement in place of the longsentences.sentences.

Solving for the Unknown.Solving for the Unknown.

In Algebra you are mostly looking for some UNKNOWN value in a given equation.In Algebra you are mostly looking for some UNKNOWN value in a given equation.

The UNKNOWN is also called a The UNKNOWN is also called a variablevariable and and may be represented by any lettermay be represented by any letter..

For example in the equation 2For example in the equation 2p p = 24,= 24,

pp is the is the unknownunknown or the or the variablevariable..

2p2p means means 22 timestimes p p (that is (that is 2 2 x x pp ) )

The number The number 22 in this case is called the in this case is called the coefficientcoefficient of p of p

So 2So 2pp=24 is the same as =24 is the same as 2 2 x x pp = 24 = 24

In this simple situation it is easy to see that In this simple situation it is easy to see that 2 2 x x 1212 = 24 so p=12 = 24 so p=12

How did we get How did we get pp to be equal to 12 ? to be equal to 12 ?

By looking for the number that will multiply 2 to give 24By looking for the number that will multiply 2 to give 24

We will get the same result by dividing both the left side and the rightWe will get the same result by dividing both the left side and the right side of the equation by 2 so that p stands alone. See? Easy.side of the equation by 2 so that p stands alone. See? Easy.

In a Math problem if you see the unknown, such as p, standing by itself it is the same In a Math problem if you see the unknown, such as p, standing by itself it is the same as as 1p1p ( that is to say ( that is to say pp==1p1p))

So So p + 2pp + 2p means means 1p + 2p 1p + 2p which is equal towhich is equal to 3p 3p

And And 3p – p3p – p = = 2p2p (Notice that 3p – p (Notice that 3p – p 3) 3)

If you have 3 apples and you give away 1 apple you will be left with 2 apples.If you have 3 apples and you give away 1 apple you will be left with 2 apples.

Each item in the equation is called a Each item in the equation is called a termterm, with the exception of the operators (+, -) , with the exception of the operators (+, -) and the equal to sign (and the equal to sign (==))..

In the equation 2p + 6 = 20, the In the equation 2p + 6 = 20, the termsterms are 2p, 6 and 20 are 2p, 6 and 20

If you add or subtract two or more unknown terms, simply add or subtract the actual If you add or subtract two or more unknown terms, simply add or subtract the actual numbers and then apply the unknown to the result:numbers and then apply the unknown to the result:

2p + 5p = 7p2p + 5p = 7p

4p -3p = p4p -3p = p

p + 5p = 6pp + 5p = 6p

3p -p = 2p3p -p = 2p

There are a couple of ways to deal with a basic There are a couple of ways to deal with a basic algebra equations in order to find the unknown.algebra equations in order to find the unknown.

One method is to continue performing the same One method is to continue performing the same actions to both sides of the equation (to the left actions to both sides of the equation (to the left and the right of the equal to sign) until you have and the right of the equal to sign) until you have the unknown terms standing on one side of the the unknown terms standing on one side of the “=“ sign and all other terms standing on the “=“ sign and all other terms standing on the opposite side of the “=“ sign.opposite side of the “=“ sign.

Performing the same actions to both sides of the Performing the same actions to both sides of the equation means that if you add, subtract, divide equation means that if you add, subtract, divide or multiply one side of the equation by a number, or multiply one side of the equation by a number, you must do the same on the opposite side of the you must do the same on the opposite side of the equation.equation.

An example will help clarify this.An example will help clarify this.

We want to find p in the equation, 2p + 7 = 21We want to find p in the equation, 2p + 7 = 21

Step 1: Decide where you would finally want the unknown, p, to stand when you Step 1: Decide where you would finally want the unknown, p, to stand when you finish solving the problem. Would you want it on the left or the right side of the ‘=‘finish solving the problem. Would you want it on the left or the right side of the ‘=‘ signsign

Step 2: Start adding and subtracting terms you would like to disappear from one side of the Step 2: Start adding and subtracting terms you would like to disappear from one side of the equation and appear on the other side until you have the unknown term standing alone.equation and appear on the other side until you have the unknown term standing alone.

Step 3: The last step normally involves dividing or multiplying each side of the equation byStep 3: The last step normally involves dividing or multiplying each side of the equation by the number associated with the unknown (or coefficient)the number associated with the unknown (or coefficient)

Now the solution:Now the solution:

2p + 7 = 212p + 7 = 21- 7 -7 - 7 -7

2p +0 = 21 -72p +0 = 21 -7

2p = 142p = 14

2p2p = = 141477

2 22 2

p = 7p = 7

What is the value of x in the following equation:What is the value of x in the following equation: 2x -6 = 15 + x2x -6 = 15 + x

Solution:Solution:

2x -6 = 15 + x 2x -6 = 15 + x <--- this is the given equation<--- this is the given equation

+6+6 +6 +6 <--- add 6 to both sides of the equation to zero out the <--- add 6 to both sides of the equation to zero out the --6 on the left6 on the left

2x -0 = 21 + x 2x -0 = 21 + x <--- after adding 6 to each side<--- after adding 6 to each side

-x-x - x - x <--- subtract x from both sides to zero out the +x on the right side<--- subtract x from both sides to zero out the +x on the right side

x = 21x = 21

(Remember that 2x –x is the same as 2x -1x which is(Remember that 2x –x is the same as 2x -1x which is equal to 1x or simply x)equal to 1x or simply x)

How difficult was that?How difficult was that?

Just keep in mind that any action you take Just keep in mind that any action you take on the on the leftleft side of the ‘=‘ sign must be side of the ‘=‘ sign must be taken taken at the same timeat the same time on the on the rightright side side of the ‘=‘. Do this before you even blink.of the ‘=‘. Do this before you even blink.

Now try the following:Now try the following: 2p -6 = 322p -6 = 32 3x + 18 = 45 - x3x + 18 = 45 - x 2p -6 = 15 + p2p -6 = 15 + p 13 +5x = 35 –x13 +5x = 35 –x 3(8 – x) +5 =2x -163(8 – x) +5 =2x -16

Another way to solve an algebra equation is by re-arranging Another way to solve an algebra equation is by re-arranging the terms in the equation so that all LIKE terms are grouped the terms in the equation so that all LIKE terms are grouped together on either side of the “together on either side of the “==“ sign.“ sign.

Before you move each term, note that the operator in front Before you move each term, note that the operator in front of a term makes the term either positive or negative.of a term makes the term either positive or negative.

Any term that you move from one side of the “Any term that you move from one side of the “==“ sign to “ sign to the opposite side will have its sign changed.the opposite side will have its sign changed.

You will re-arrange the equation by moving terms to join You will re-arrange the equation by moving terms to join their likes on either side of the “their likes on either side of the “==“ sign.“ sign.

The unknown terms together with their coefficients are The unknown terms together with their coefficients are considered like or similar terms. Move them to one side of considered like or similar terms. Move them to one side of the equal to sign.the equal to sign.

All the other terms (without the variables) belong to a All the other terms (without the variables) belong to a different group of LIKE terms. Move them to the opposite different group of LIKE terms. Move them to the opposite side of the equal to signside of the equal to sign

2p + 72p + 7 == 2121

Grouping Like terms together:Grouping Like terms together:2p 2p == 2121 – 7– 7

You notice that You notice that + + 77 has become has become -7-7 as soon as it as soon as itcrossed the “crossed the “==“ sign from the left side to the right“ sign from the left side to the rightside.side.

2p = 142p = 14

2p2p = = 14142 22 2

p = 7p = 7

You can normally carry out the rearrangement in a You can normally carry out the rearrangement in a single step but it is okay to use as many steps to single step but it is okay to use as many steps to group LIKE terms together as you feel comfortable:group LIKE terms together as you feel comfortable:

Let us find the value of p in the equation Let us find the value of p in the equation 2p - 6 2p - 6 == 15 + p 15 + p

Solution:Solution: 2p - 6 2p - 6 == 15 + p 15 + p

Grouping like terms together:Grouping like terms together: 2p –p 2p –p == 15 + 6 15 + 6 p p == 21 21

Try the following:Try the following: 2x - 6 2x - 6 == 32 32 3p + 18 3p + 18 = = p + 45p + 45 2p -6 2p -6 == 27 + p 27 + p 13 + 5p 13 + 5p == 35 –p 35 –p 3(8 – x) +5 3(8 – x) +5 ==2x -162x -16

RATIOS & PROPORTIONSRATIOS & PROPORTIONS

A ratio compares two or more A ratio compares two or more actual quantitiesactual quantities using smaller using smaller equivalent quantitiesequivalent quantities or numbers. or numbers.

For example if we are told that a basket contains 20 apples and For example if we are told that a basket contains 20 apples and 30 oranges, we can represent these actual quantities using a 30 oranges, we can represent these actual quantities using a ratio by saying that the ratio of ratio by saying that the ratio of apples to orangesapples to oranges is 20 to 30. is 20 to 30.

We can also write the ratio of apples to oranges as 20:30We can also write the ratio of apples to oranges as 20:30

Using smaller equivalent numbers we can simplify 20:30 and Using smaller equivalent numbers we can simplify 20:30 and represent the ratio as 2:3 or 2 to 3represent the ratio as 2:3 or 2 to 3

From the question it is clear that there is a combined total of From the question it is clear that there is a combined total of 50 (apples + oranges) in the basket50 (apples + oranges) in the basket

The previous question could have been asked in a different way:The previous question could have been asked in a different way:

A basket contains apples and oranges in the ratio of 2 apples to 3 oranges and there is a A basket contains apples and oranges in the ratio of 2 apples to 3 oranges and there is a total of 50 apples and oranges together. How many of each fruit is in the basket?total of 50 apples and oranges together. How many of each fruit is in the basket?

In this case we are given the smaller equivalent numbers or apples and oranges and are In this case we are given the smaller equivalent numbers or apples and oranges and are required to find out the actual quantities of each fruit in the basketrequired to find out the actual quantities of each fruit in the basket

We can write the ratio of apples to oranges as We can write the ratio of apples to oranges as 22::33 or or 22 toto 33

This means that if we decide to group the 50 fruits in equal quantities using the given ratios This means that if we decide to group the 50 fruits in equal quantities using the given ratios then for every 5 (i.e. then for every 5 (i.e. 22++33) fruits there will be ) fruits there will be 22 apples and apples and 33 oranges. oranges.

22++3 3 = 5 is the = 5 is the total ratiototal ratio

In other words 2 out of 5 (two-fifth) In other words 2 out of 5 (two-fifth) OfOf the total number of fruits are apples and 3 out of 5 the total number of fruits are apples and 3 out of 5 (three-fifth) (three-fifth) OfOf the total number of fruits are oranges. the total number of fruits are oranges.

In fractions:In fractions:

2 2 OfOf 50 are apples 50 are apples5 5

3 3 OfOf 50 are oranges 50 are oranges55

When a given quantity ,Q, is split in the ratio a:b:c, we can find the When a given quantity ,Q, is split in the ratio a:b:c, we can find the actual quantities of a, b and c by writing the fractions for the actual quantities of a, b and c by writing the fractions for the ratios.ratios.

If we represent the actual quantities for a, b and c with the If we represent the actual quantities for a, b and c with the symbols symbols

A, B and C respectively then we can calculate as follows:A, B and C respectively then we can calculate as follows:

A = A = a a xx Q Q a+b+c a+b+c

B = B = b b xx Q Q a+b+ca+b+c

C = C = c c xx Q Q a+b+ca+b+c

The above formulas do not need to be memorized. They are only The above formulas do not need to be memorized. They are only intended to be understood and applied in ratio and proportions intended to be understood and applied in ratio and proportions calculations.calculations.

Another ExampleAnother Example There are 350 students in a school. The ratio of boys to girls in the school is There are 350 students in a school. The ratio of boys to girls in the school is

5 to 2. What are the actual numbers of boys and girls in the school.5 to 2. What are the actual numbers of boys and girls in the school.

Solution:Solution:

Total number of students = 350Total number of students = 350

Ratio of Boys to Girls = Ratio of Boys to Girls = 55::22

Number of Boys=Number of Boys= 55 x x 350 = 350 = 55 xx 350 3505050

55 + + 22 7 711

= 250= 250

Number of Girls=Number of Girls= 22 x x 350 = 350 = 22 xx 350 3505050

55 + + 22 7 711

= 100= 100

Now Try TheseNow Try These There are 18 girls in a class. If there are six more There are 18 girls in a class. If there are six more

boys than girls in the class, find the ratio of boys boys than girls in the class, find the ratio of boys to girls in the class. What is the total number of to girls in the class. What is the total number of students in the class?students in the class?

A box contains 24 pencils and 42 pens. What is A box contains 24 pencils and 42 pens. What is the ratio of pens to pencils in the box?the ratio of pens to pencils in the box?

David, Kim and Isaiah want to share an amount of David, Kim and Isaiah want to share an amount of $120 in the ratio 2:3:5. How much will each $120 in the ratio 2:3:5. How much will each person get?person get?