We know how to express some numbers as squares and cubes.

**Exponents
and Powers**

We know how
to express some numbers as squares and cubes. For example, we write 5^{2}
for 25 and 5^{3} for 125.

In general
terms, an expression that represents repeated multiplication of the same factor
is called a **power**.
The number 5 is called the **base** and the number 2 is called the **exponent**
(more often called as power). The exponent corresponds to the number of times the
base is used as a factor.

** **

__1. Powers
with positive exponents__

Value of
powers given by positive whole number exponents quite often increase rapidly.

Observe the
following example:

2^{1}
=2

2^{2}
=
2×2=4

2^{3}
=
2×2×2=8

2^{4}
=
2×2×2×2
=16

2^{5}
=
2×2×2×2×2=32

2^{6}
=
2×2×2×2×2×2=64

2^{7}
=
2×2×2×2×2×2×2=128

2^{8}
=
2×2×2×2×2×2×2×2=256

2^{9}
=
2×2×2×2×2×2×2×2×2=512

2^{10}
=
2×2×2×2×2×2×2×2×2×2=1024

At this rate
of increase, what do you think 2^{100} will be?

In fact,
**2 ^{100}=1267650600228229401496703205376**

Thus, we
understand that the positive exponential notation with positive power could be useful
when we come across with large numbers.

** **

__2. Powers
with zero and negative exponents__

Observe this
pattern:

**2 ^{5}= 32**

**2 ^{4}= 16**

**2 ^{3}= 8**

**2 ^{2}= 4**

**2 ^{1}= 2**

**2 ^{0}=?**

Starting
from the beginning, what happens in the successive steps? We find that the result
is half of that of the previous step. So, what can we say about 2^{0} ?
If we prepare a table like this for 3^{5}, 3^{4}, 3^{3},
and so on what will it tell us about 3^{0}? We can use the same process
as in this pattern, to conclude that any non-zero number raised to the zero exponent
must result in 1. Thus,

* a ^{0} = *

Let us see
what happens if we extend the above pattern further downward.

As before,
starting from the beginning, in the successive steps, we find that the result is
half of that of the previous step. Since 2^{0} = 1, the next step is 2^{–1},
whose value is the previous step’s value 1, divided by 2, that is 1/2 . Next is
2^{–}^{2}, which is the same as 1/2 divided by 2, that is 1/4 and so on. Thus,

**2 ^{3} = 8**

**2 ^{2} = 4**

**2 ^{1} = 2**

**2 ^{0} = 1**

**2 ^{–1} = **

**2 ^{–2} = **

**2 ^{–3} = **

In general, *a*^{−}^{m}*=*** 1/a^{m}** , where

** **

__3. Expanded
form of numbers using exponents__

In the lower
classes, we have learnt how to write a whole number in the expanded form. For example,
5832 = 5×1000
+ 8×100
+ 3×10
+ 2×1

= 5×10^{3}
+ 8×10^{2}
+ 3×10^{1}
+ 2 (when we use exponential notation).

What shall
we do if we get decimal places? Powers of 10 with negative exponents come to our
rescue!

Thus, 58.32
= 50 + 8 + 3/10 + 2/100

= 5×10 + 8×1 + 3× (1/10) + 2
× (1/100)

= 5×10^{1}
+ 8×10^{0}
+ 3×10^{–1}
+ 2×10^{–2}

**Try these**

Expand the following
numbers using exponents:

1.8120 2. 20305
3. 3652.01 4. 9426.521

**Solution:**

**(i) 8120 =** (8 × 1000) + (1 × 100) + (2 × 10) + 0 × l

= (8 × 10^{3}) +
(1 × 10^{2}) + (2 × 10^{1})

**(2) 20305 =** (2 × 10000) + (0 × 1000) + (3 × 100) + ( 0 × 10) + (5 × 1)

= (2 × 10^{4}) +
(3 × 10^{2}) + 5

**(3) 3652.01** = 3000 + 600 + 50 + 2 + 0/10 + 1/100

= (3 × 1000) + (6 × 100) + (5 × 10) + (2 × 1) + 1 × 1/100)

= (3 × 10^{3}) +
(6 × 10^{2}) + (5 × 10^{1}) + 2 + (1 × 10^{−2})

**(4) 9426.521** = (9 × 1000) + (4 × 100) + (2 × 10) + (6 × 1) + (5/10) +
(2/100) + (1/1000)

= (9 × 10^{3}) + (4 × 10^{2}) + (2 × 10^{1})
+ 6 + (5 × 10^{−1}) + (2 × 10^{−2}) + (1 × 10^{−3})

__4. Laws
of Exponents__

Laws of exponents
arise out of certain basic ideas. A **positive exponent** of a number indicate how many
times we use that number in a multiplication whereas a **negative** **exponent **suggests us how many times we use that number in a division, since the opposite
of** **multiplying is dividing.

__• Product
law__

According
to this law, when multiplying two powers that have the same base, we can add the
exponents. That is,

**a ^{m} × a^{n}
= a^{m + n}**

where *a* (*a*≠0),
*m*, *n* are integers. Note that the base should be the same in both the quantities.

__Examples:__

__• Quotient
law__

According
to this law, when dividing two powers that have the same base we can subtract the
exponents. That is,

**a ^{m} / a^{n}
= a^{m-n}**

^{}

where *a* (*a*
≠
0), *m*, *n* are integers. Note that the base should be the same in both the quantities.

How does
it work? Study the following examples.

__Examples:__

__• ____Power law__

According
to this law, when raising a power to another power, we can just multiply the exponents.

*(***a ^{m})^{n}
= a^{mn}**

^{}

where *a* (*a*
≠
0), *m*, *n* are integers.

__Examples:__

[(-
2)^{3} ]^{2} = (- 2)^{3×2} by law and (-2)^{6}
= 64

[(-2)^{3}]^{2}
= [(-2)×
(-2)×
(-2)]^{2}
= [-8]^{2}
= 64

**Try these**

Verify the following
rules (as we did above). Here, *a,b* are
non-zero integers and m, n are any integers.

1. Product of same powers to power of product rule: *a* * ^{m}*
×

2. Quotient of same powers to power of quotient rule: *a ^{n}*/

3. Zero exponent rule: *a*^{0}
= 1.

**Verification:**

Let *a* = 2; *b* = 3; *m* = 2

1. *a** ^{m}*
×

2. *a** ^{m}*
/

3. *a*^{0} = 2^{0} = 1.

__Example 1.36__

__Example 1.37__

Simplify
and write the answer in exponential form:

(i) (3^{5}
÷ 3^{8})^{5} × 3^{–5} (ii) ( −3)^{4} × (5/3)^{4}

**Solution:**

(i) (3^{5} / 3^{8})^{5}
× 3^{–5} = (3^{5-8}) × 3^{-5} = (3^{-3})^{5}
× 3^{-5} = 3^{-3×5} = 3^{-3×5} = 3^{-15} × 3^{-5}
= 3^{-15-5} = 3^{-20}

(ii) (-3)^{4} × (5/3)^{4} = 3^{4}
× 5^{4}/3^{4} = 5^{4}

__Example 1.38__

Find *x* so that ( −7 ) ^{x}^{+}^{2} × ( −7 )^{5} = ( −7)^{10}^{}

**Solution:**

(-7 )^{x}^{+}^{2} × (-7)^{5} = (-7)^{10}

(-7) ^{x}^{+2+5 }= (7)^{10}^{}

Since the
bases are equal, we equate the exponents to get

* x*
+ 7 = 10

* **x =*10 - 7 = 3

** **

__5. Standard
Form and Scientific Notation__

Standard
form of a number is just the number as we normally write it. We use expanded notation
to show the value of each digit. That is, it is exhibited as a sum of each digit
duly multiplied by its matching place value (like ones, tens, hundreds etc.,). For
example, **195
**is in standard form. It can be expanded as** 195 = 1 × 100 +
9 × 10 + 5 × 1**.

Astronomers,
biologists, engineers, physicists and many others come across quantities whose measures
require very small or very large numbers. If they write the numbers in standard
form, it may not help us to understand or make computations easily. **Scientific notation**
is a way to make these numbers easier to work with.

To write
in scientific notation, follow the form **S** × **10^{a}**

__Examples:__

**Some more examples:**

(a) The diameter
of the earth is 12756000 *miles*. This can
be easily written in scientific form as 1.2756×10^{7} *miles*.

(b) The volume
of Jupiter is about 143300000000000 km^{3.} This can be easily written in
scientific form as 1.433×10^{14} km^{3}.

(c) The size
of a bacterium is 0.00000085 mm. This can be easily written in scientific form as
8.5 ×10^{−}^{7} mm

**Note**

1. The positive exponent in 1.3 ×10^{12} indicates that it is a large number.

2. The negative exponent in 7.89 × 10^{–21} indicates that it is a small
number.

__Example 1.39__

Combine the scientific notations: (i)
(7 × 10^{2})(5.2 × 10^{7}) (ii) (3.7 × 10^{-5})(2
× 10^{-3})^{}

**Solution:**

(i) (7 ×
10^{2})(5.2 × 10^{7}) = 36.4 × 10^{9} = 3.64 × 10^{10}

(ii) (3.7
× 10^{-5}) (2 × 10^{-3}) =7.4×10^{–8}

__Example 1.40__

Write the
following scientific notations in standard form:

(i) 2.27
× 10^{-4} (ii) Light travels at 1.86 ×10^{5} *miles* per second.

**Solution:**

(i) 2.27
× 10^{-4} = 0.000227.

(ii) Light
travels at 1.86 ×10^{5} *miles* per second = 186000 *miles*
per second

**Try these**

1. Write in standard
form: Mass of planet Uranus is 8.68 × 10^{25} *kg*.

**Solution:**

Mass of Planet Uranus = 86800000000000000000000000 kg

2. Write in scientific
notation: (i) 0.000012005 (ii) 4312.345 (iii) 0.10524 (iv)The distance between the
Sun and the planet Saturn 1.4335×10^{12} *miles*.

**Solution:**

(i) 0.000012005 = 1.2005 × 10^{−5}

(ii) 4312.345 = 4.312345 × 10^{3}

(iii) 0.l 524 = 1.0524 × 10^{−1}

(iv) The distance between Sun and the planet Saturn is 1.4335 × 10^{12}
miles

Tags : Numbers | Chapter 1 | 8th Maths , 8th Maths : Chapter 1 : Numbers

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8th Maths : Chapter 1 : Numbers : Exponents and Powers | Numbers | Chapter 1 | 8th Maths

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