Question

Find the square of 45.

Answer 1

Hint: We will first express the given number as a difference of two numbers. We will then apply a suitable algebraic identity to simplify the expression. We will further solve the equation to get the required answer. The square of a number is obtained by multiplying a number by itself.

Formula used:
 \[{(a - b)^2} = {a^2} - 2ab + {b^2}\]

Complete step-by-step answer:
We are supposed to find the square of 45.
The square of any number can be obtained either by usual multiplication or by the application of appropriate identities.
To find the square of 45, we will apply the identity \[{(a - b)^2} = {a^2} - 2ab + {b^2}\].
For this, we need to write 45 as a difference of two numbers.
We can write 45 as \[45 = 50 - 5\].
So, \[a - b = 50 - 5\].
Here, \[a = 50\] and \[b = 5\].
Let us first find \[{a^2}\] i.e., \[{50^2}\].
\[{a^2} = {50^2} = 50 \times 50\]
Multiplying the terms, we get
\[ \Rightarrow {a^2} = 2500\] ……….\[(1)\]
Now, we will find \[2ab\]. So,
\[2ab = 2 \times 50 \times 5 = 500\] ……….\[(2)\]
Finally, let us find \[{b^2}\] i.e., \[{5^2}\].
\[{b^2} = {5^2} = 5 \times 5\]
Multiplying the terms, we get
\[ \Rightarrow {b^2} = 25\]……….\[(3)\]
Substituting equations \[(1)\], \[(2)\], and \[(3)\] in the identity \[{(a - b)^2} = {a^2} - 2ab + {b^2}\], we get
\[{45^2} = 2500 - 500 + 25\]
Adding and subtracting the terms, we get
\[ \Rightarrow {45^2} = 2025\]
Hence, the square of 45 is 2025.

Note: We can also solve the above problem by using the identity \[{(a + b)^2} = {a^2} + 2ab + {b^2}\].
To apply this, we will express 45 as a sum of two numbers.
We can write \[45\] as \[45 = 40 + 5\].
So, \[a + b = 40 + 5\].
Here, \[a = 40\] and \[b = 5\].
Let us first find \[{a^2}\] i.e., \[{40^2}\].
\[{a^2} = {40^2} = 40 \times 40\]
Multiplying the terms, we get
\[ \Rightarrow {a^2} = 1600\] ………. \[(4)\]
Now, we will find \[2ab\]. So,
\[2ab = 2 \times 40 \times 5 = 400\] ………. \[(5)\]
Finally, let us find \[{b^2}\] i.e., \[{5^2}\].
\[{b^2} = {5^2} = 5 \times 5\]
Multiplying the terms, we get
\[ \Rightarrow {b^2} = 25\]………. \[(6)\]
Substituting equations \[(4)\], \[(5)\], and \[(6)\] in the identity \[{(a + b)^2} = {a^2} + 2ab + {b^2}\], we get
\[{45^2} = 1600 + 400 + 25\]
Adding the terms, we get
\[ \Rightarrow {45^2} = 2025\]
Hence, the square of 45 is 2025.