• Exponent
• Roman Numerals
• Divisibility Rules
• Factors
• Factorial
• LCM
• GCF
• Modular Arithmetic
• Scientific Notations

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Product form of numbers

Large numbers broken down into smaller forms is known as product form of numbers.

To find the product form of numbers let's say 4⁷
  4⁷ is written in the product form as 4⁷= 4 × 4 × 4 × 4 × 4 × 4 × 4

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Expressing the product form of numbers by exponents

To express product from of the numbers by exponents we count that how many times the number multiplied by itself. That is the exponent of that number.

To find the exponent form of numbers let's say 32.
 Write the product form of 32. 32= 2 x 2 x 2 x 2 x 2= 2⁵.
 Here 2 is multiplied by 5 times with itself. So that exponent is 5 and base is given 2.
 We get the exponent form is 2⁵.

Simple Example

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Square of a Number

A square of number is the product of some integer with itself.
For example, 25 is a square number, since it can be written as 5 × 5.

Note:

•  To find a square of number we multiply the number two times by itself.
•  For example square of 4 can be represented as "4×4=16". We can express it in exponent form as 4^2.
•  Square of negative number is positive number.

Simple Example

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Examples square of variables:

• 1. ab² = a × b × b
• 2. (ab)² = ab × ab
• 3. (ab)^-2 = -(ab) × (-ab)= (ab)²

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Cube of a Number

A cube of number is just the number multiplied by 3 times with itself.

For example, to find cube of 3.
 Cube of 3 is 3 times 3.
 Cube of 3 = 3 × 3 × 3= 27.
 We can express it in exponent form as 3³

Note:

•  Cube of negative number is negative number.

Simple Example

Examples cube of variables:

• 1. ab³= a × b × b x b
• 2. (ab)³= ab×ab×ab

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What is the value of numbers in exponent form?

We can find the value of numbers given in exponent form by multiplying the base with itself as many times the exponent is given.

To find the value of 3⁵.
 3⁵ mean multiplying the 3 with itself 5 times.
 3⁵= 3 × 3 × 3 × 3 × 3
 3 × 3 × 3 × 3 × 3= 243.

Exponent : Word Problems

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Exponent rules

Any number with an exponent of 0 (except 0) is equals to 1. So that x⁰ = 1.
Examples: 5⁰= 1, 15⁰= 1, 123⁰= 1, 154⁰= 1, 120⁰= 1

When base raised to the positive power value; x¹= x
Examples: (10)¹= 10, (15)¹= 15, (124)¹= 124 and (-10)¹= -10, (-15)¹= -15, (-124)¹= -124

Simple Example

When base raised to the negative power value; x^-1 = 1/x.
Examples: 5^-6= 1/5⁶, 4^-3= 1/4³, 7^-5= 1/7⁵

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Volume of water

Word Problem:

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Rule for multiplication when the base is the same

When we multiply the numbers having the same base, the base of the product remains the same and exponents are added to get the index of the product.

For example: 8⁴× 8⁵⁼ 8⁴⁺⁵=8⁹

Note:

•  If a is any rational number and m and n are any positive integers then a^m× aⁿ = a^{m + n}

Simple Example

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Medium Example:

(7/5)³×(7/5)⁸
= (7/5)³⁺⁸
= (7/5)¹¹

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Advance Example:

(y)⁴×(y)^6
= (y)^4+6
= (y)¹⁰

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Real Life Example : Word Problem

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Rule for division when the base is the same

If a is any rational number other than 0 and m and n are any positive integers such that m>n, then a^m÷ aⁿ = a^m-n

Note:

•  When we divide the numbers having the same base, the base remains the same and exponents are subtracted.
•  When the index of the dividend is greater than that of divisor, subtract the exponent of the divisor from that of the dividend.

Simple Example

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Medium Example:

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Advance Example:

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Advance Example;

(a) ⁴×(a) ⁶
=1/(a) ^6-4
=1/(a) ²

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Real Life Example

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Rule for index of a number in index form

If x is any rational number and m and n are positive integers then
(x^m )ⁿ= (x)^{m × n}

If we say, (2³ )⁵
 (2³ )⁵
 Multiply the index.
 (2³ )⁵= (2)^{3 × 5}= 2¹⁵

Simple Example

(a⁴ )⁵
(a⁴ )⁵
= a^{4 × 5}
= a²⁰

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Medium Example

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Rule for the index of a product

If x and y are any rational number and m is any positive integers then
(x× y) ^m = x^m × y^m .

If we say, (4×3)⁷.
 (4×3)⁷= (4)⁷× (3)⁷

Simple Example

(a×b)⁵
(a×b)⁵ = (a)⁵× (b)⁵

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Medium Example:

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Advance Example:

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Rule for index of a number in the numerator-denominator form

If x / y is any rational number and m is any positive integers then
(x/ y) ^m = x^m/ y^m

Let us say,
 (3/ 5) ² = 3²/ 5² and (x/y) ² = x²/ y²

Simple Example

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Medium Example:

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Complementary examples with more rules together

Advance Example:

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Fractional index

The square of number is shown by writing the number with exponent 2.
Index ½ represents the square root of the number.
These are known as fractional indices.

Let us say square of 4, we will write as 4².
Now, if we want to write square root of 4, we will write as 4^1/2.
As well as the cube root of 8 is written as 8^1/3

Note:

• The nth root of a number is shown by the index 1/n.

How to read the numbers as fractional indices?

See the following examples:

 9^1/2 means the square root of 9.
 1000^1/3 means the cube root of 1000.
 625^1/4 means the fourth root of 625.
 1000^1/10 means the square root of 1000.

Simple Example

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Medium Example:

Find the values of the following index numbers.

•  125^1/3= Cube root of 125 = 5
•   144^1/2= Square root of 144 = 12
•   216^1/3= Cube root of 216 = 6
•   169^1/2= Square root of 169 = 13

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Multiplication fractional indices

As you know If a is any rational number and m and n are any positive integers then
a^m× aⁿ = a^m+n
The same rule is applied for fractional indices when we multiply the numbers as index form with the same base.

If we say, 7^1/2× 7^1/3
 Add the index numbers. 7^1/2× 7^1/3 = 7^{1/2+ 1/3}
 Simplify added index form. 7^{1/2+ 1/3} = 7^3+2/6
 7^3+2/6= 7^5/6 means the 5^th power of the 6^th root of 7.

Simple Example

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Division fractional indices

If a is any rational number other than 0 and m and n are any positive integers such that m>n, then a^m÷ aⁿ = a^m-n

The same rule is applied for fractional indices when we multiply the numbers as index form with the same base.
If we say,
7^1/3÷ 7^1/3
 Subtract the index numbers. 7^1/3÷ 7^1/3= 7^{1/3 - 1/3}
 Simplify index form. = 7^{1/3 - 1/3} = 7⁰
 Write the value. 7⁰ = 1

Simple Example

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Rule for index of a number for fractional indices

If x is any rational number and m and n are positive integers then
(x^m )ⁿ= x^{m × n}

If we say^, (2^1/3 )^1/5
  Multiply the index numbers. (2^1/3 )^1/5 = 2^{1/3 ×1/5}
  Simplify index form. 2^{1/3 ×1/5} = 2^1/15
  Write the value. 2^1/15

Simple Example

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Rule for the index of a product for fractional indices

If x and y are any rational number and m is any positive integers then
(x× y) ^m = x^m × y^m .

We can say,
(5×3)^1/7= (5) ^1/7× (3) ^1/7
Or (a×b) ^1/5= (a) ^1/5× (b) ^1/5

Simple Example

Simplify (5 × 9) 1/3
(5×9) ^1/3
= (5) ^1/3× (9) ^1/3

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Medium Example:

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Complementary examples with more rules together

Simple Example:

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Advance Example:

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• Addition
• Subtraction
• Multiplication
• Division
• Integers
• Natural numbers
• Even odd