Write down the units digits of the squares of the following numbers
$ (i)24 $
$ (ii)78 $
$ (iii)35 $
Answer
176.4k+ views
Hint: When the numbers are small we can easily calculate and write the answer, but when the number is more than 30 it becomes difficult to remember the squares and cubes. That’s when we start using the algebraic expressions and formulae to calculate the value of squares and cubes. For this particular sum we would be using the expression $ {(a + b)^2} = {a^2} + {b^2} + 2ab $ .
Complete step-by-step answer:
$ (i) - 24 $
Using the $ {(a + b)^2} = {a^2} + {b^2} + 2ab $ , we will have to define $ a\& b $ .
We can further split $ 24 $ as $ 20 + 4 $ . This is because it is easy to find squares of numbers which have 0 at units place.
After the split we can define $ a = 20 $ and $ b = 4 $
Now using the expression we get
$ \Rightarrow {(24)^2} = {(20)^2} + {(4)^2} + 2 \times 4 \times 20 $
$ \Rightarrow {(24)^2} = 400 + 16 + 160 $
$ \Rightarrow {(24)^2} = 576............(1) $
Thus the Units digit for $ {24^2} $ is $ 6 $ from $ Equation1 $ .
VERIFICATION:
In order to verify we can just take the units digit of the given number and square it. The units’ digit value obtained after squaring the unit digit of the original number is our final answer. In our case we will have to take a unit digit of $ 24 $ i.e. $ 4 $ .
Square of $ 4 $ is $ 16 $ . Thus the units digit of $ 16 $ is $ 6 $ .
$ \therefore $ Units digit of $ {24^2} $ is $ 6 $
Verified.
So, the correct answer is “6”.
$ (ii) - 78 $
Using the $ {(a + b)^2} = {a^2} + {b^2} + 2ab $ , we will have to define $ a\& b $ .
We can further split $ 78 $ as $ 70 + 8 $ . This is because it is easy to find squares of numbers which have 0 at units place.
After the split we can define $ a = 70 $ and $ b = 8 $
Now using the expression we get
$ \Rightarrow {(78)^2} = {(70)^2} + {(8)^2} + 2 \times 8 \times 70 $
$ \Rightarrow {(78)^2} = 4900 + 64 + 1120 $
$ \Rightarrow {(78)^2} = 6084............(2) $
Thus the Units digit for $ {78^2} $ is $ 4 $ from $ Equation2 $ .
VERIFICATION:
In order to verify we can just take the units digit of the given number and square it. The units’ digit value obtained after squaring the unit digit of the original number is our final answer. In our case we will have to take a unit digit of $ 78 $ i.e. $ 8 $ .
Square of $ 8 $ is $ 64 $ . Thus the units digit of $ 64 $ is $ 4 $ .
$ \therefore $ Units digit of $ {78^2} $ is $ 4 $
Verified.
So, the correct answer is “4”.
$ (iii) 35 $
Using the $ {(a + b)^2} = {a^2} + {b^2} + 2ab $ , we will have to define $ a\& b $ .
We can further split $ 35 $ as $ 30 + 5 $ . This is because it is easy to find squares of numbers which have 0 at units place.
After the split we can define $ a = 30 $ and $ b = 5 $
Now using the expression we get
$ \Rightarrow {(35)^2} = {(30)^2} + {(5)^2} + 2 \times 5 \times 30 $
$ \Rightarrow {(35)^2} = 900 + 25 + 300 $
$ \Rightarrow {(35)^2} = 1225............(3) $
Thus the Units digit for $ {35^2} $ is $ 5 $ from $ Equation3 $ .
VERIFICATION:
In order to verify we can just take the units digit of the given number and square it. The units’ digit value obtained after squaring the unit digit of the original number is our final answer. In our case we will have to take a unit digit of $ 35 $ i.e. $ 5 $ .
Square of $ 5 $ is $ 25 $ . Thus the units digit of $ 25 $ is $ 5 $ .
$ \therefore $ Units digit of $ {35^2} $ is $ 5 $
Verified
So, the correct answer is “5”.
Note: In such a type of numerical it is always advisable to verify the given answer with the shortcut as illustrated. Also Students have to memorize the algebraic expressions by heart so that it becomes easier to solve much more complex numerical. It is highly beneficial not only in solving the given numericals but also in chapters like Ratio & Proportion, Complex Numbers.
Complete step-by-step answer:
$ (i) - 24 $
Using the $ {(a + b)^2} = {a^2} + {b^2} + 2ab $ , we will have to define $ a\& b $ .
We can further split $ 24 $ as $ 20 + 4 $ . This is because it is easy to find squares of numbers which have 0 at units place.
After the split we can define $ a = 20 $ and $ b = 4 $
Now using the expression we get
$ \Rightarrow {(24)^2} = {(20)^2} + {(4)^2} + 2 \times 4 \times 20 $
$ \Rightarrow {(24)^2} = 400 + 16 + 160 $
$ \Rightarrow {(24)^2} = 576............(1) $
Thus the Units digit for $ {24^2} $ is $ 6 $ from $ Equation1 $ .
VERIFICATION:
In order to verify we can just take the units digit of the given number and square it. The units’ digit value obtained after squaring the unit digit of the original number is our final answer. In our case we will have to take a unit digit of $ 24 $ i.e. $ 4 $ .
Square of $ 4 $ is $ 16 $ . Thus the units digit of $ 16 $ is $ 6 $ .
$ \therefore $ Units digit of $ {24^2} $ is $ 6 $
Verified.
So, the correct answer is “6”.
$ (ii) - 78 $
Using the $ {(a + b)^2} = {a^2} + {b^2} + 2ab $ , we will have to define $ a\& b $ .
We can further split $ 78 $ as $ 70 + 8 $ . This is because it is easy to find squares of numbers which have 0 at units place.
After the split we can define $ a = 70 $ and $ b = 8 $
Now using the expression we get
$ \Rightarrow {(78)^2} = {(70)^2} + {(8)^2} + 2 \times 8 \times 70 $
$ \Rightarrow {(78)^2} = 4900 + 64 + 1120 $
$ \Rightarrow {(78)^2} = 6084............(2) $
Thus the Units digit for $ {78^2} $ is $ 4 $ from $ Equation2 $ .
VERIFICATION:
In order to verify we can just take the units digit of the given number and square it. The units’ digit value obtained after squaring the unit digit of the original number is our final answer. In our case we will have to take a unit digit of $ 78 $ i.e. $ 8 $ .
Square of $ 8 $ is $ 64 $ . Thus the units digit of $ 64 $ is $ 4 $ .
$ \therefore $ Units digit of $ {78^2} $ is $ 4 $
Verified.
So, the correct answer is “4”.
$ (iii) 35 $
Using the $ {(a + b)^2} = {a^2} + {b^2} + 2ab $ , we will have to define $ a\& b $ .
We can further split $ 35 $ as $ 30 + 5 $ . This is because it is easy to find squares of numbers which have 0 at units place.
After the split we can define $ a = 30 $ and $ b = 5 $
Now using the expression we get
$ \Rightarrow {(35)^2} = {(30)^2} + {(5)^2} + 2 \times 5 \times 30 $
$ \Rightarrow {(35)^2} = 900 + 25 + 300 $
$ \Rightarrow {(35)^2} = 1225............(3) $
Thus the Units digit for $ {35^2} $ is $ 5 $ from $ Equation3 $ .
VERIFICATION:
In order to verify we can just take the units digit of the given number and square it. The units’ digit value obtained after squaring the unit digit of the original number is our final answer. In our case we will have to take a unit digit of $ 35 $ i.e. $ 5 $ .
Square of $ 5 $ is $ 25 $ . Thus the units digit of $ 25 $ is $ 5 $ .
$ \therefore $ Units digit of $ {35^2} $ is $ 5 $
Verified
So, the correct answer is “5”.
Note: In such a type of numerical it is always advisable to verify the given answer with the shortcut as illustrated. Also Students have to memorize the algebraic expressions by heart so that it becomes easier to solve much more complex numerical. It is highly beneficial not only in solving the given numericals but also in chapters like Ratio & Proportion, Complex Numbers.
Recently Updated Pages
If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

The coordinates of the points A and B are a0 and a0 class 11 maths JEE_Main

Trending doubts
What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?
