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Q) 53 *56?
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Q) -72?
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Q) 81; base -3?
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Q) 1460000=146* 10000=146 * 104?
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Q) (2/5)3?
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Q) For any two non-zero integers a, b, and any positive integer m?
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Q) For any two non-zero integers a and any integer n?
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Q) Express 288 as the product of exponents through prime factorization?
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Q) Write each number as the power of a given base -343; base -7?
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Q) Evaluate (-4)5?
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Q) Evaluate (2/5)5?
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Q) Evaluate 16; base 2?
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Q) Evaluate 81; base -3?
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Q) Write the following numbers in the expanded form? 29870, 56840, 78943, 56340.
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Q) Express the following numbers in expanded form? (a) 6* 1010 + 5* 10 4 + 0 * 102 + 5 * 100. (b) 6* 123 + 3* 5 4 + 0 * 16 2 + 8 * 100.
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Q) Say true or false and justify your answer? (i) 10 * 10 12=10012. (ii) 23>52.
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Q) Simplify and express each of the following in exponential form? (i)(23 * 34*4)/(3*42). (ii) ((5,sup>2)3854)+57.
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Q) Express each of the following as a product of prime factors only in exponential form? (i) 106*165.
(ii) 365.
(ii) 246*986.
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Q) What is Decimal Number System?
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Q) Express 32/243 in power notation?
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Q) Very large number are difficult to read, understand, compare and operate upon. To make all these easier,we use exponent, converting many of the large numbers in a shorter form?
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Q) (a)33300000=333* 100000=333 * 105? (b)36789090=777* 1244500=777 * 102?
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Q) (a)81; base -8?
(b) -33; base 7?
(c) -443; base -9?
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Q) let us calculate? (i)( 3 a5 b3 c ) 2 (ii) ( 3 a2 b12 c4)2.
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Q) We can express it using powers of 10 in the exponent form?
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Q) What is Laws Of Exponents and give an examples?
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Q) What is Powers and give an examples?
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- Exponents
- Multiplying powers with same base
- Laws Of Exponents
- Powers
- Decimal Number System
- Expressing Large Numbers In The Standard Form
- Examples On Product Form
- Very large numbers are difficult to read, understand, compare and operate upon. To make all these easier, we use exponents, converting many of the large numbers in a shorter form.
- The following are exponential forms of some numbers?
`10,000 = 10^4`(read as 10 raised to 4)
`243 = 3^5`, 128 = 27.
Here, 10, 3 and 2 are the bases, whereas 4, 5 and 7 are their respective exponents. We also say, 10,000 is the 4th power of 10, 243 is the 5th power of 3, etc. - Numbers in exponential form obey certain laws, which are:
For any non-zero integers a and b and whole numbers m and n,
- `a^m xx a^n= a^(m + n)`
- `a^m ÷ a^n= a^(m – n)`, m > n
- `(a^m)^n= a^(mn)`
- `a^m xx b^m = (ab)^m`
- `a^m ÷ b^m =(a/b)^m `
- `a^0= 1`
- (–1)even number = 1
(–1)odd number = – 1
The laws of exponents are explained here along with their examples.
1. Multiplying powers with same base
For example: x2 `xx` x3, 23 `xx` 25, (-3)2 `xx` (-3)4
In multiplication of exponents if the bases are same then we need to add the exponents
Consider the following:
1. `2^3 xx 2^2= (2 xx 2 xx 2) xx (2 xx 2) = 2^3 + 2^3 = 25`
2. `3^4 xx 3^2 = (3 xx 3 xx 3 xx 3) xx (3 xx 3) = 3^4 + 2 = 36`
3. `(-3)^3 xx (-3)^3 = [(-3) xx (-3) xx (-3)] xx [(-3) xx (-3) xx (-3) xx (-3)]`
= `(-3)^3 + 4 = (-3)^7`
4. `m^5 xx m^3 = (m xx m xx m xx m xx m) xx (m xx m xx m) = m^5 + m^3 = m^8`
From the above examples, we can generalize that during multiplication when the bases are same then the exponents are added.
`a^m xx a^n = a^(m+n)`
In other words, if 'a' is a non-zero integer or a non-zero rational number and m and n are positive integers, then
`a^m xx a^n = a^(m + n)`
Similarly, `(a/b)^m xx (a/b)^n = (a/b)^(m + n)`
- Exponents can be added only when the bases are same.
- Exponents cannot be added if the bases are not same like
`m^5 xx n^7, 2^3 xx 3^4`
1.` 5^3 xx5^6`
= `(5 xx 5 xx 5) xx (5 xx 5 xx 5 xx 5 xx 5 xx 5)`
= `5^3+6`[here the exponents are added]
= 59
2. `(-7)^10 xx (-7)^12`
= `[(-7) xx (-7) xx (-7) xx (-7)xx (-7) xx (-7) xx (-7) xx (-7) xx (-7) xx (-7)]xx [( -7) xx (-7) xx`
`(-7)xx (-7) xx (-7)xx (-7) xx (-7) xx (-7) xx (-7) xx (-7) xx (-7) xx (-7)]`.
= `(-7)^10+12` [exponents are added]
= `(-7)^22`
3. `(1/2)^4 xx ( 1/2)^3 `
=`[(1/2) xx ( 1/2) xx ( 1/2) xx ( 1/2)] xx [ ( 1/2) xx ( 1/2) xx ( 1/2)]`
`=(1/2)^(4 + 3)`
`=(1/2)^7`
Example
Formula
am x an = am+n
Example
Problem 1.
(1). let us calculate
Formula
am ÷ an = am - n
Example
Problem 2.
(2). let us calculate
`(m^2+n^4)/(m^5+n^3)`
Solution:
`(m^2+n^4)/(m^5+n^3)`
`(m^2+n^4)/(m^5+n^3)=m^(2-5)n^(4-3)`
=`m^-3n^1`
=`n/m^3`
Formula
(am)n= amn
Example
Problem 3.
(3) . let us calculate `( 3 a^5 b^3 c )^2`
solution:
`( 3 a^5 b^3 c )^2`
=`3^2 xx (a^5)^2 xx (b^3)^2 xx c^2`
=`9xxa^10xxb^6xxc^2`
=`9a^10b^6c^2`
A power contains two parts exponent and base.
We know `2 xx 2 xx 2 xx 2 = 2^4`, where 2 is called the base and 4 is called the power or exponent or index of 2.
| words | multiplication | power | value |
|---|---|---|---|
| 4 to the first power | `4^1` | 4 | |
| 4 to the second power | `4^2` | 16 | |
| 4 to the third power | `4^3` | 64 | |
| 4 to the fourth power | `4^4` | 256 | |
| 4 to the fifth power | `4^5` | 1024 |
Examples on evaluating powers (exponents):
1. Evaluate each expression:
(i) 54.
Solution:
- 54
= 5 · 5 · 5 · 5 → Use 5 as a factor 4 times.
= 625 → Multiply.
(ii) (-3)3.
Solution:
= (-3) · (-3) · (-3) → Use -3 as a factor 3 times.
= -27 → Multiply.
(iii) -72.
Solution:
- -72
= -`(7^2)` → The power is only for 7 not for negative 7
= -(7 · 7) → Use 7 as a factor 2 times.
= -(49) → Multiply.
= -49
(iv) `(2/5)^3`
Solution:
- `(2/5)^3`
=` (2/5) xx (2/5) xx (2/5) ` → Use `2/5` as a factor 3 times.
= `8/(125)` → Multiply the fractions
2. Write each number as the power of a given base:
(a) 16; base 2
Solution:
- 16; base 2
Express 16 as an exponential form where base is 2
The product of four 2's is 16.
Therefore, 16
= 2 · 2 · 2 · 2 = 24
Therefore, required form = `2^4`
(b) 81; base -3
Solution:
- 81; base -3
Express 81 as an exponential form where base is -3
The product of four (-3)'s is 81.
Therefore, 81
= (-3) · (-3) · (-3) · (-3)
= (-3)4
Therefore, required form = `(-3)^4`
(c) -343; base -7
Solution:
- -343; base -7
Express -343 as an exponential form where base is -7
The product of three (-7)'s is -343.
Therefore, -343
= (-7) · (-7) · (-7)
= `(-7)^3`
Therefore, required form = `(-7)^3`
Where m and n are whole numbers and m < n;
We can generalize that if 'a' is a non-zero integer or a non-zero rational number and m and n are positive integers, such that m > n, then
`a^m` ÷ `a^n = a^(m - n)` if m < n, then `a^m` ÷ `a^n = (1/a)^(n - m)`
Similarly, `(a/b)^m` ÷ `(a/b)^n = (a/b)^(m -n)`
For example:
1. `7^(10) ÷ 7^(10)` = `7^(10)/7^(10)`
= `(7 xx 7 xx 7 xx 7 xx 7 xx 7 xx 7 xx 7 xx 7 xx 7)/( (7 xx 7 xx 7 xx 7 xx 7 xx 7 xx 7 xx 7 xx 7 xx 7)`
= 710 - 10[here exponents are subtracted]
= = 7 0 =1
we can express it using powers of 10 in the exponent form:
In exponents we will mainly learn about the exponential form and product form, negative integral exponents, positive and negative rational exponents, laws of exponents etc,. To write large numbers in shorter form, so that it becomes very convenient to read, understand and compare, we use exponents. In exponents we will learn more about exponents and their uses.
When a number is multiplied with itself a number of times, then it can be expressed as a number raised to the power of a natural number, equal to the number of times the number is multiplied with itself.
| words | multiplication | power | value |
|---|---|---|---|
| 4 to the first power | `4^1` | 4 | |
| 4 to the second power | `4^2` | 16 | |
| 4 to the third power | `4^3` | 64 | |
| 4 to the fourth power | `4^4` | 256 | |
| 4 to the fifth power | `4^5` | 1024 |
For example:
`3 xx 3 xx 3 xx 3 xx 3` can be written as 35 and is read as 3 raised to the power 5. Here the base is 3 and the exponent is 5.
Similarly, for any rational number 'a' and a positive integer, we define an as `a xx a xx a xx a xx ...... a` (n times)
For example:
- `(-2)^4 = -2 xx -2 xx -2 xx -2`
- `(-2)^3 = -2 xx -2 xx -2`
`a^n` is called the nth power of a and can be read as:
a raised to the power n.
The rational number a is called the base and n is called the exponent or the power or the index.
Therefore the notation of writing the product of rational number by itself several times is called the exponential notation or the power notation. Exponential notation is also known as power notation.
- We can write `-5 xx -5 xx -5 xx -5` in the exponential form as (-5)4 and is read as -5 raised to the power 4. Here, (-5) is the base.
- Also, `3/2 xx 3/2 xx 3/2 xx 3/2 xx 3/2` in the exponential form is written as `(3/2)^5` and is read as `3/2` raised to the power 5. Here, `3/2` is the base, 5 is the exponent.
- We can write 53 in the product form as 5 * 5 * 5 and its product as 125.
- Similarly, `(-4/3)^2` is written as `-4/3 xx -4/3` and its product is `(16)/9`.
Powers with positive and negative exponents such as 52 or (-5)2 is the positive exponent and 5-2or (-5)-2 is the negative exponent.
We know that `10^2= 10 xx 10 = 100`
`10^0 = 1`
`10^1 = 10`
`10^2= 10 xx 10 = 100`
`10^3= 10 xx 10 xx 10 = 1000`
`10^4= 10 xx 10 xx 10 xx 10 = 10000`
`10^5= 10 xx 10 xx 10 xx 10 xx 10 = 100000`
Powers with negative exponents is also known as negative integral exponents.
So, `10^-1 = 1/(10)`
`10^-2 = 1/10^2`
`10^-3 = 1/10^3`
Thus, for any non zero rational number 'a' and a positive integer we define.
`a^-n = 1/a^n`
i.e., `a^-n` is the reciprocal of `a^n` or `a^[-n]` is the multiplicative inverse of `a^n`
1. `1460000=146 xx 10000=146 xx 10^4`
=`14.6 xx 10000=14.6 xx 10^5`
=`1.46 xx 10000=1.46 xx 10^6`
2. `33300000=333 xx 10000=333 xx 10^5`
=`33.3 xx 100000=33.3 xx 10^6`
=`3.33 xx 100000=3.33 xx 10^7`