EXPONENTS AND SCIENTIFIC NOTATION
Exponents and Scientific Notation :
Here we are going to see the basic concept of exponents and scientific notation.
What is exponents?
The exponent of a number says, how many times the number has to be multiplied by itself.
For example,
53 = 5 x 5 x 5
In words, 53 could be called as 5 to the power 3 or 5 cube.
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102 is called as 10 to the power 2 or simply called as 10 square. |
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103 is called as 10 to the power 3 or simply called as 10 cube. |
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If we have 1/2 in the power, we can simply write the base inside the radical or square root. If we have 1/3 in the power,we can simply write the base inside the cube root |
How to move an exponents or powers to the other side ?
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If the power goes from one side of equal sign to the other side, it will become its reciprocal. That is x = 42. |
The other names of exponent are index and power.
Laws of Exponents
Law 1 :
xm ⋅ xn = xm+n
Example :
34 ⋅ 35 = 34+5
34 ⋅ 35 = 39
Law 2 :
xm ÷ xn = xm-n
Example :
37 ÷ 35 = 37-5
37 ÷ 35 = 32
Law 3 :
(xm)n = xmn
Example :
(32)4 = 3(2)(4)
(32)4 = 38
Law 4 :
(xy)m = xm ⋅ ym
Example :
(3 ⋅ 5)2 = 32 ⋅ 52
(3 ⋅ 5)2 = 9 ⋅ 25
(3 ⋅ 5)2 = 225.
Law 5 :
(x / y)m = xm / ym
Example :
(3 / 5)2 = 32 / 52
(3 / 5)2 = 9 / 25
Law 6 :
x-m = 1 / xm
Example :
3-2 = 1 / 32
3-2 = 1 / 9
Law 7 :
x0 = 1
Example :
30 = 1
Law 8 :
x1 = x
Example :
31 = 3
Law 9 :
xm/n = y -----> x = yn/m
Example :
x1/2 = 3
x = 32/1
x = 32
x = 9
Law 10 :
(x / y)-m = (y / x)m
Example :
(5 / 3)-2 = (3 / 5)2
(5 / 3)-2 = 32 / 52
(5 / 3)-2 = 9 / 25
Law 11 :
ax = ay -----> x = y
Example :
3m = 35 -----> m = 3
Law 12 :
xa = ya -----> x = y
Example :
k3 = 53 -----> k = 5
Scientific Notation Rules
Every number in the scientific notation must be in the form of
a x 10n
where 1 ≤ a < 10 and n must be a positive or negative integer.
To convert a number into scientific notation, first we have to identify where the decimal point and non zero digit come.
There are two cases in it.
Case 1 :
To move the decimal point to the left, we have to count number of digits as explained in the example given below.
According to the example given above, we have to move the decimal point 3 digits to the left and exponent of 10 should be 3 (positive integer)
When we do so, we get the scientific notation of the given number.
Hence, 2301.8 = 2.3018 x 10³
Case 2 :
To move the decimal point to the right, we have to count number of digits as explained in the example given below.
According to the example given above, we have to move the decimal point 5 digits to the right and exponent of 10 should be -5 (negative integer)
When we do so, we get the scientific notation of the given number.
Hence, 0.000023 = 2.3 x 10⁻⁵
Important Note:
If we don't find decimal point at anywhere of the given number, we have to assume that there is decimal point at the end of the number.
For example,
2300000 -------------> 2300000.
Here, the non zero digit comes first and decimal point comes next. So we have to apply case 1 to convert this number into scientific notation.
If you would like to have practice problems on exponents,
If you would like to have practice problems on scientific notation,
After having gone through the stuff above, we hope that the students would have understood the basic stuff about exponents and scientific notation.
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WORD PROBLEMS
Word problems on simple equations
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Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
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OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
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Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits
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