An exponent is the number of times that a certain number is multiplied by itself. It is written as a small-sized number at the top right of the number.
In the example above, `5^2` is read as "5 to the power of 2" or "5 raised to 2".
Large numbers broken down into smaller forms is known as product form of numbers.
To find the product form of numbers let's say 4^{7}
4^{7} is written in the product form as 4^{7}= 4 × 4 × 4 × 4 × 4 × 4 × 4
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= 2^{5}.
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 2^{5}.
Let us find the prime factors of 128.
128= 2 x 2 x 2 x 2 x 2 x 2 x 2 = 2^{7}
Here 2 is multiplied by 7 times with itself. So that exponent is 7 and base is given
2.
We get the exponent form is 2^{7}.
Let us find the prime factors of 32.
32= 2 x 2 x 2 x 2= 2^{4}
Here 2 is multiplied by 5 times with itself. So that exponent is 4 and base is given 2.
We get the exponent form is 2^{4}.
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.
Square of 35.
= (-35)^{2}
= (-35) × (-35)
= 1225
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 3^{3}
Find the cube of 25.
= (-25)^{3}
= (-25) × (-25) × (-25)
= -15625
To find the value of 3^{5}.We can find the value of numbers given in exponent form by multiplying the base with itself as many times the exponent is given.
The group numbers of boxes are 3. In every group there are 4 boxes. So that base
is 4 and exponent is 3. We can write this in exponent form as 4^{3}.
The product form of 4^{3}= 4×4×4
We can read 4^{3 }as 4 raised to 3.
The value of 4^{3}= 4×4×4= 64
The group numbers of stars are 4. In every group there are 12 stars. So that base
is 12 and exponent is 4. We can write this in exponent form as 12^{4}.
The product form of 12^{4}= 12 × 12 × 12 × 12
We can read 12^{4 }as 12 raised to 4.
The value of 12^{4}= 12 × 12 × 12 × 12= 20736
Any number with an exponent of 0 (except 0) is equals to 1. So that x^{0} = 1.
Examples: 5^{0}= 1, 15^{0}= 1, 123^{0}= 1, 154^{0}=
1, 120^{0}= 1
When base raised to the positive power value; x^{1}= x
Examples: (10)^{1}= 10, (15)^{1}= 15, (124)^{1}= 124 and
(-10)^{1}= -10, (-15)^{1}= -15, (-124)^{1}= -124
(2/3)^{6}
Here 2/3 is the base and 6 is exponent.
For (2/3)^{6} mean 2/3 is 6 times. So that,
(2/3)^{6}= 2/3× 2/3 × 2/3× 2/3 × 2/3
×2/3= 2 × 2 × 2 × 2 × 2 × 2 / 3 × 3 × 3 × 3 × 3 × 3= 64/729
When base raised to the negative power value; x^{-1} = 1/x.
Examples: 5^{-6}= 1/5^{6}, 4^{-3}= 1/4^{3}, 7^{-5}=
1/7^{5}
(12)^{-4}
Here 12 is the base and -4 is exponent.
For (12)^{-4}mean 1/ (12)^{4}
(12)^{-4}=1/ (12)^{4}
=1/12 × 1/12 × 1/12 × 1/12= 1 × 1 × 1 × 1 / 12 × 12 × 12 × 12 = 1/ 20736
Volume of a cube= (side)^{3}
Volume of a cube container = 9^{3}= 9×9×9= 729.
Volume of a cube container is 729 cubic inches.
The volume of kerosene in cubic inches
= 729 /3= 243 cubic inches
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: 8^{4}× 8^{5= }8^{4+5}=8^{9}
7^{5}×7^{4}
^{= (7 × 7 × 7 × 7 × 7)×( 7× 7 × 7 × 7)}
^{= (7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7)}
^{= the number 7, (5 + 4) times}=7^{9}
^{So that, }7^{5}×7^{4}=7^{9}
(7/5)^{3}×(7/5)^{8}
=^{ }(7/5)^{3+8}
= (7/5)^{11}
(y)^{4}×(y)^{6}
^{= }(y)^{4+6}
^{= }(y)^{10}
There are 81 ways to complete the project.
There are 3^{2} ways to complete the 2 documents in Part A. There are 3^{2}
ways complete the 2 documents in Part B. The number of ways to complete the project
all 4 documents is the product of 3^{2}and 3^{2}
The product of 3^{2}and 3^{2}
= 3^{2×}3^{2}
= 3^{2+2}
= 3^{4}
= 81
If a is any rational number other than 0 and m and n are any positive integers such that m>n, then a^{m}÷ a^{n }= a^{m-n}
= 8^{9-4 }
= 8^{5}
8^{4}÷ 8^{9 }^{}
= 1/8^{9-4}
^{}= 1/8^{5}
6^{8}÷6^{4}
^{= (6×6×6×6×6×6×6×6)} ÷^{ ( 6×6×6×6)}
^{= ( 6×6×6×6)}
^{= 1296}
^{= }(2/3)^{8}^{-3}^{}
=(2/3)^{5}
= (2/3)× (2/3)× (2/3)× (2/3)× (2/3)
= 2×2×2×2×2 / 3×3×3×3×3
= 32/243
(a)^{ 4}×(a)^{ 6}
=1/(a)^{ 6-4}
=1/(a)^{ 2}
To finding more women were doing jobs from home, we have to divide these
two expressions shown for women and men. That is 7^{6 }and 7^{2}
7^{6}/7^{2}
= 7^{6-2}
= 7^{4}
^{=7×7×7×7}
^{= 2401}
If we say, (2^{3 })^{5}If x is any rational number and m and n are positive integers then
(x^{m })^{n}= (x)^{m × n}
(a^{4 })^{5}
(a^{4 })^{5}
= a^{4 × 5}
= a^{20}
[(-3/4)^{2}]^{2}
We will multiply the index numbers.
= (-3/4)^{2×2}
= (-3/4)^{4}
= -3× -3× -3× -3/ 4× 4 × 4× 4
= 81/256
If we say, (4×3)^{7}.If x and y are any rational number and m is any positive integers then
(x× y)^{ m }= x^{m }× y^{m }.
(a×b)^{5}
(a×b)^{5} = (a)^{5}× (b)^{5}
(5×2)^{3}
= (5)^{3}× (2)^{3}
= (5×5×5) ×(2×2×2)
= 1000
( 1/3× 2/5 ) ^{2}
= (1/3)^{2 }× (2/5 )^{2}
= 1×1×2×2/ 3×3×5×5^{}
= 4/225
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 }= 3^{2}/ 5^{2 }and (x/y)^{
2 }= x^{2}/ y^{2}
(5/2)^{4}
= (5)^{4}/ (2)^{4}
= (5×5×5×5) / (2×2×2×2)
= 625/16
(-6/5)^{2}
= (-6)^{2}/ (5)^{2}
= (-6×-6) / (5×5)
= 36/25
(5 × 2)^{3 }÷ (5 × 2)^{1}
We will first subtract the index here.
= (5×2)^{ 3}^{-1}
= (5×2)^{2}
= (5)^{2}× (2)^{2}
= 5 × 5 × 2 × 2
= 100
We will first add the index here.
= (5^{1 })^{2+3}
= (5^{1 })^{5}
= 5^{1×5}
=5^{ 5}
= 3125
(a × b)^{ 3 }÷ (a× b)^{ 5}
= 1/ (a × b)^{ 5}^{-3}
= 1/ (a × b)^{ 2}
= 1/ (2 × 3)^{ }^{2 }put the^{ }value for a=2 and b = 3
= 1/ (2)^{2}× (3)^{2}
1/ 2 × 2 × 3 × 3
=1/36^{}
(a)^{ 0 }÷ (b)^{ 0}
= (15)^{ 0 }÷ (17)^{ 0} put the^{ }value for a=15 and b
= 17
= 1÷ 1
=1
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 4^{2}.
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}
How to read the numbers as fractional indices?
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.
4 × 4 × 4 × 4 = 256.
Hence, 4^{4 }= 256.
Conversely, 4 is the fourth root of 256.
16 × 16 = 256.
Hence, 16^{2 }= 256.
Conversely, 16 is the square root of 256.
Find the values of the following index numbers.
As you know If a is any rational number and m and n are any positive integers then
a^{m}× a^{n }= 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.^{}
9^{1/3}× 9^{1/3}
= 9^{1/3+ 1/3}
= 9^{1+1/3}
= 9^{2/3}
14^{1/4}× 14^{1/5}
= 14^{1/4+ 1/5}
= 14^{5+4/20}
= 14^{9/20}
^{x1/3}× x^{1/5}
= x^{1/3+ 1/5}
= x^{5+3/15}
= x^{8/15}
If a is any rational number other than 0 and m and n are any positive integers such that m>n, then a^{m}÷ a^{n }= 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} = 7^{0}
Write the value. 7^{0} = 1
13^{1/4}÷ 13^{1/5}
= 13^{1/4- 1/5}
= 13^{5-4/20}
= 13^{1/20}
13^{1/20} means the 20^{th} root of 13.
m^{1/3}÷ m^{1/5}
= m^{1/3 - 1/5}
= m^{5-3/15}
= m^{2/15}
8^{1/4}÷ 8^{1/6}
= 8^{1/4 - 1/6}
= 8^{ 3-2/12}
= 8^{ 1/12}
If we say^{, }(2^{1/3 })^{1/5 }If x is any rational number and m and n are positive integers then
(x^{m })^{n}= x^{m × n}
(25)^{ 1/2 ×1/2}
Multiply the index numbers.
= 25^{1/4}
8^{ 1/2 ×1/9}
Multiply the index numbers.
= 8^{1/18}
y^{ 1/2 × 1/3}
Multiply the index numbers.
= y^{1/6}
We can say,If x and y are any rational number and m is any positive integers then
(x× y)^{ m }= x^{m }× y^{m }.
Simplify (5 × 9) 1/3
(5×9)^{ 1/3}
= (5)^{ 1/}^{3}× (9)^{ 1/}^{3}
(25×8)^{ 1/}^{3}^{}
= (25)^{ 1/}^{3}× (8)^{ 1/}^{3}
= 5 × 2
= 10
(100 × 64)^{ 1/}^{2}^{}
= (100)^{ 1/}^{2}× (64)^{ }^{1/}^{2}
= 10 × 8
= 80
(625 × 256)^{ 1/}^{2} ÷ (625 × 256)^{ 1/}^{4}^{}
= (625 × 256)^{ 1/}^{2}^{ - 1/}^{4}^{}
= (625 × 256)^{ 1/}^{4}^{ }
= (625)^{ 1/}^{4}^{ }× (256)^{ 1/}^{4}
= 5 × 4
= 20
(625 × 256)^{ 1/}^{2} ÷ (625 × 256)^{ 1/}^{4}^{}
[(81)^{1 }]^{1/2} × [(81)^{1 }]^{1/4}
= [(81)^{1 }]^{1/2} ^{+ 1/4}
= (81^{ })^{3/4}
Means the 3^{nd }power of the 4^{th} root of 81.
= 27
(a× b)^{ }^{1/}^{3 }÷ (a× b)^{ }^{1/}^{5
}put the^{ }value for a=2 and b = 3
= (2× 3)^{ }^{1/}^{3 }÷ (2× 3)^{ }^{1/}^{5}
= (2× 3)^{ }^{1/3- }^{1/5}
= (2× 3)^{ }^{5-3/15}
= (2× 3)^{ 2/15}
= 2^{2/15}× 3^{ 2/15}
For any given number make pairs...
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