Manipulating with exponent is very interesting and funny too.

**Unit Digit of Numbers in Exponential Form**

Manipulating with exponent is very interesting
and funny too.

We know that 9^{3 }=
9 ×
9 ×
9 =
729 , thus the unit digit (the last number of expanded form) of 9^{3} is
9. Similarly, 4^{4} is 4 × 4 × 4 × 4 = 256 . Thus the unit digit of 4^{4}
is 6.

Can you guess the unit digit of 230^{116}
, 181^{47} , 55^{4} , 56^{20} and 9^{29} ?

It is very difficult to find by expanding
the exponential form. But, we can try to tell the unit digit by observing some patterns.

Look at the following number pattern.

10^{3}
=
10 ×10
×10
=
1000

10^{4}
=
10 ×10
×10
×10
=
10000

10^{5}
=
10 ×10
×10
×10
×10
=
100000

Thus, multiplying 10 by itself several
times, we always get the unit digit as 0. In
other words, 10 raised to the power of any number has the unit digit 0. That is, the unit digit of 10* ^{x}*
is always 0, for any positive integer

This is also true when the base is multiples
of 10. Consider,

40^{2}
=
(4 ×10)^{2} = 4^{2} ×10^{2}

=16 ×100
=
1600

Similarly, 230^{116}
=
(23 ×10)^{116} = 23^{116} ×10^{116}

Thus, the unit digit of 230^{116} is 0.

Now, observe that,

1^{5} =1×1×1×1×1=1

1^{6} =1×1×1×1×1×1=1

We learnt to expand 11 as 10+1.

So,( 10 +**1**)^{2} =1**1**^{2}
=1**1**×1**1** = 12**1**

Similarly, 13**1**=130 +**1**= (13×10)+**1**

[ (13 ×10) +**1** ]^{2} = 13**1**^{2}
= 13**1**
×13**1**
= 1716**1**

Hence, if the exponential number is in
the form 1* ^{x}* or [(multiple of 10) +1]

Therefore, the unit digit of 181^{47} is 1.

Similarly, by observing the following
patterns, we can conclude that the unit digit of number with base ending with 5
is 5 and number with base ending with 6 is 6.

5^{1}=5

5^{2} =5×5=25

5^{3} =25×5=125

6^{1}=6

6^{2} =6×6=36

6^{3} =36×6=216

Therefore, the unit digit of 5**5**^{4}
=
(
50 +
**5**)^{4} is **5** and the unit digit of 5**6**^{20}
=
(
50 +
**6**)^{20} is **6**.

We conclude that, for the base number
whose unit digits are 0,1,5 and 6 the unit digit of a number corresponding to any
positive exponent remains unchanged.

__Example 3.11 __

Find the unit digit of the
following exponential numbers:

(i) 25^{23}

(ii) 81^{100}

(iii)
46^{31}

**Solution**

(i) 25^{23}

Unit digit of base 25 is 5 and power
is 23

Thus, the unit digit of 25^{23}
is 5.

(ii) 81^{100}

Unit digit of base 81 is 1 and power
is 100

Thus, the unit digit of 81^{100}
is 1.

(iii) 46^{31}

Unit digit of base 46 is 6 and power
is 31

Thus, the unit digit of 46^{31}
is 6.

**Try these**

**Find the unit digit of
the following exponential numbers:**

**(i) 106 ^{21}
(ii) 25^{8} (iii) 31^{18} (iv) 20^{10}**

**i) 106 ^{21}**

The unit digit of 106^{21} is 6.

**ii) 25 ^{8}**

The unit digit of 25^{8} is 5.

**iii) 31 ^{18}**

The unit digit of 31^{18} is 1.

**iv) 20 ^{10}**

The unit digit of 20^{10 }is 0.

Look into the following example. Observe
the pattern of unit digit when the base is 4.

4^{1}=4 (odd power) 4^{2}=4×4=16 (even power)

4^{3 }= 4 × 4 × 4 = 16 × 4 = 64 (odd power)
4^{4 }=64×4=256 (even power)

4^{5} = 256×4 =1024 (odd power) 4^{6 }= 1024×4=4096 (even power)

Note that for base ending with 4, the
unit digit of the expanded form alternates between 4 and 6. Further we can notice
when the power is odd its unit digit is 4 and when the power is even
it is 6.

Similarly, when the base unit is 9,

9^{1}=9 (odd power) 9^{2 }=9×9=81 (even power)

9^{3} = 9 × 9 × 9 = 81 × 9 = 729 (odd power)
9^{4
}=729×9=6561 (even power)

9^{5} = 6561× 9 = 59049 (odd power) 9^{6} = 59049 × 9 = 531441 (even power)

Thus, for base ending with 9, the unit
digit after the expansion is 9 for odd power and is 1
for even power.

As we have seen in the earlier case,
this rule is applicable, when the base is in the form of [(multiple of 10)+4] or
[(multiple of 10)+9].

For example consider, 24^{12}

In this, unit digit of base 24 is 4 and
power is 12 (even power).

Therefore, unit digit of 24^{12}
is 6.

Similarly consider, 89^{21}

Here unit digit of base 89 is 9 and power
is 21 (odd power).

Therefore, unit digit of 89^{21}
is 9.

We conclude that, for base ending with
4, the unit digit of the expanded form is 4 for odd power and is 6 for even power.
Similarly, for base ending with 9, the unit digit of the expanded form is 9 for
odd power and is 1 for even power. Remember, 4 and 6 are complements of 10. Also,
9 and 1 are complements of 10.

__Example 3.12__

** **Find the unit digit of the large numbers:

(i)** **4^{7}

(ii) 64^{10}

**Solution**

(i) 4^{7}

Unit digit of base 4 is 4 and power is
7 (odd power).

Therefore, unit digit of 4^{7}
is 4.

(ii) 64^{10}

Unit digit of base 64 is 4 and power
is 10 (even power).

Therefore, unit digit of 64^{10}
is 6.

**Try these**

**Find the unit digit of the following exponential numbers:**

**(i) 64 ^{11}**

Power 11 (odd power) so,
the unit digit is 4.^{}

**(ii) 29 ^{18}**

Power 18 (even power) so,
the unit digit is 1.^{}

**(iii) 79 ^{19} **

Power 19 (odd power) so, the
unit digit is 9.^{}

**(iv) 104 ^{32}**

Power 32 (even power) so,
the unit digit is 6.^{}

**Example 3.13**

** **Find the unit digit of the large numbers: (i)** **9^{12 }(ii) 49^{17}

**Solution**

(i) 9^{12}

Unit digit of base 9 is 9 and power is
12 (even power).

Therefore, unit digit of 9^{12}
is 1 .

(ii) 49^{17}

Unit digit of base 49 is 9 and power
is 17 (odd power).

Therefore, unit digit of 49^{17}
is 9.

The following activity will help you
to find the unit digits of exponent numbers whose base is ending with 2,3,7 and
8.

**Activity**

Observe the table given
below. The numbers in first column, that is 2,3,7 and 8 denotes the unit digit of
base of the given exponent number and the numbers in the first row, that is 1,2,3
and 0 stands for the remainder when power is divided by 4.

For example, consider 2^{6}

Unit digit of base 2 is
**2** and power is 6. When the power 6 is divided by 4, we get the remainder
as 2.

From the table, we see
that **2** and 2 corresponds to 4. Therefore, the unit digit of 2^{6}
is 4. To verify, 2^{6} = 2 × 2 × 2 × 2 × 2 × 2 = 64 .

Similarly, consider 117^{20}

Unit digit of base 117
is **7** and power is 20. When the power 20 is divided by 4, we get the remainder
as 0.

From the table, we see
that **7** and 0 corresponds to 1. Therefore, the unit digit of 117^{20}
is 1.

Now, you can extend this
activity for finding the unit digits of any exponent number whose base is ending
with any of 2,3,7 or 8.

Tags : Algebra | Term 2 Chapter 3 | 7th Maths , 7th Maths : Term 2 Unit 3 : Algebra

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7th Maths : Term 2 Unit 3 : Algebra : Unit Digit of Numbers in Exponential Form | Algebra | Term 2 Chapter 3 | 7th Maths

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