**Basic Math Rules : **

In this section, we are going to learn some basic rules in math which will be much useful for the middle grade students to become master in math.

**Rule 1 :**

Positive + Positive = Add

The result will be positive

Example :

2 + 1 = 3

**Rule 2 :**

Negative + Negative = Add

The result will be negative

Example :

-3 + (-5) = -8

**Rule 1 : **

Negative + Positive = Subtract

Take sign of number with largest absolute value

Example :

-3 + 5 = 2

**Rule 2 : **

Positive + Negative = Subtract

Take sign of number with largest absolute value

Example :

3 + (-5) = -2

**Rule 1 : **

Positive x Positive = Positive

Example :

3 x 5 = 15

**Rule 2 : **

Negative x Negative = Positive

Example :

(-3) x (-5) = 15

**Rule 2 : **

Positive x Negative = Negative

Example :

3 x (-5) = -15

**Rule 2 : **

Negative x Positive = Negative

Example :

-3 x 5 = -15

**Rule 1 : **

Positive ÷ Positive = Positive

Example :

20 ÷ 4 = 5

**Rule 2 : **

Negative ÷ Negative = Positive

Example :

(-20) ÷ (-4) = 5

**Rule 2 : **

Positive ÷ Negative = Negative

Example :

20 ÷ (-4) = -5

**Rule 2 : **

Negative ÷ Positive = Negative

Example :

-20 ÷ 4 = -5

**Rule 1 : **

x^{m} ⋅ x^{n} = x^{m+n}

Example :

3^{4} ⋅ 3^{5} = 3^{4+5}

3^{4} ⋅ 3^{5} = 3^{9}

**Rule 2 : **

x^{m} ÷ x^{n} = x^{m-n}

Example :

3^{7} ÷ 3^{5} = 3^{7-5}

3^{7} ÷ 3^{5} = 3^{2}

**Rule 3 : **

(x^{m})^{n} = x^{mn}

Example :

(3^{2})^{4} = 3^{(2)(4)}

(3^{2})^{4} = 3^{8}

**Rule 4 : **

(xy)^{m} = x^{m} ⋅ y^{m}

Example :

(3 ⋅ 5)^{2} = 3^{2} ⋅ 5^{2}

(3 ⋅ 5)^{2} = 9 ⋅ 25

(3 ⋅ 5)^{2} = 225.

**Rule 5 : **

(x / y)^{m} = x^{m} / y^{m}

Example :

(3 / 5)^{2} = 3^{2} / 5^{2}

(3 / 5)^{2} = 9 / 25

**Rule 6 : **

x^{-m} = 1 / x^{m}

Example :

3^{-2} = 1 / 3^{2}

3^{-2} = 1 / 9

**Rule 7 : **

x^{0} = 1

Example :

3^{0} = 1

**Rule 8 : **

x^{1} = x

Example :

3^{1} = 3

**Rule 9 : **

x^{m/n} = y -----> x = y^{n/m}

Example :

x^{1/2} = 3

x = 3^{2/1}

x = 3^{2}

x = 9

**Rule 10 : **

(x / y)^{-m} = (y / x)^{m}

Example :

(5 / 3)^{-2} = (3 / 5)^{2}

(5 / 3)^{-2} = 3^{2} / 5^{2}

(5 / 3)^{-2} = 9 / 25

**Rule 11 : **

a^{x} = a^{y} -----> x = y

Example :

3^{m} = 3^{5} -----> m = 3

**Rule 12 : **

x^{a} = y^{a} -----> x = y

Example :

k^{3} = 5^{3} -----> k = 5

This rule can be used to simplify or evaluate complicated numerical expressions with more than one binary operation easily.

Very simply way to remember PEMDAS rule :

** P -----> Parenthesis**

** E -----> Exponents **

** M -----> Multiplication**

** D ****-----> **Division

** A ****-----> **Addition

** S ****-----> **Subtraction

**Important notes :**

1. In a particular simplification, if you have both multiplication and division, do the operations one by one in the order from left to right.

2. Multiplication does not always come before division. We have to do one by one in the order from left to right.

3. In a particular simplification, if you have both addition and subtraction, do the operations one by one in the order from left to right.

**Examples : **

**16 ÷ 4 x 3 = 4 x 3 = 12**

**18 - 3 + 6 = 15 + 6 = 21 **

In the above simplification, we have both division and multiplication. From left to right, we have division first and multiplication next.

So we do division first and multiplication next.

For more examples on PEMDAS, **please click here**

The formula given below can be used to find the percent increase or decrease of a value.

The change may be an increase or a decrease.

Here, the original amount is the value before increase or decrease.

For more examples on percent increase / decrease, **please click here**

Place value of a digit in a number is the digit multiplied by thousand or hundred or whatever place it is situated.

**For example,**

In 2__5__486, the place value of 5 is

= 5 ⋅ 1000

= 5000

Here, to get the place value of 5, we multiply 5 by 1000.

Because 5 is at thousands place.

Face value of a digit in a number is the digit itself.

More clearly, face value of a digit always remains same irrespective of the position where it is located.

**For example,**

In 2__5__486, the face value of 5 is 5.

The difference between place value and face has been illustrated in the picture shown below.

Acute Angle : Less than 90°

Obtuse Angle : More than 90°

Right Angle : 90°

Straight Angle : 180°

**Complementary Angles : **

Two angles the sum of whose measures is 90 degrees.

**Supplementary Angles : **

Two angles the sum of whose measures is 180 degrees.

**Triangles : **

1. The sum of the lengths of any two sides of a triangle is greater than the third side.

2. The sum of all the three angles of a triangle is 180°.

**Isosceles Triangle : **

Two sides equal ; two equal angles

**Equilateral Triangle : **

Three sides equal ; three equal angles

**Right Triangles :**

Pythagorean Theorem :

a^{2} + b^{2} = c^{2}

where a and b are the measures of the legs of the triangle and c is the hypotenuse.

**Mean (Average) :**

Sum of all values divided by number of values.

**Median : **

Middle value when the values are arranged numerically.

**Mode : **

The data value that occurs most frequently.

**Probability of the event A : **

P(A) = The frequency of A / Total sample size

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