Question

Find the square of the following number: 46

Answer 1

Hint: To solve the given question, we will first find out the definition of squaring and we will see how to square any number. Then, we will expand the number and write it in form of the sum of the two numbers obtained by expanding the given number and then we will square it and apply the identity ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$ and hence get the square.

Complete step-by-step answer:
Before we start to solve the given question we will first find out what we mean when we say a square of a number. A square of a number is defined as the multiplication of the number with the same number. Thus, we can say that we will get the square of a number when we multiply that number with the same number. If n is a number, then its square is denoted by \[{{n}^{2}}.\] The square of a number, \[{{n}^{2}}=n\times n.\]

We can write the square of 46 as follows by expanding it the number as a sum of $40$ and $6$:
\[{{\left( 46 \right)}^{2}}={{\left( 40+6 \right)}^{2}}\]\[\Rightarrow {{\left( 46 \right)}^{2}}={{\left[ 4\left( 10 \right)+6 \right]}^{2}}\]
We know that according to the identity: ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$ , therefore:
\[\Rightarrow {{\left( 46 \right)}^{2}}={{\left[ 4\left( 10 \right) \right]}^{2}}+{{\left[ 6 \right]}^{2}}+2\left( 4 \right)\left( 6 \right)\left( 10 \right)\]
\[\Rightarrow {{\left( 46 \right)}^{2}}={{4}^{2}}\times {{10}^{2}}+{{6}^{2}}+2\times 4\times 6\times 10\]
\[\Rightarrow {{\left( 46 \right)}^{2}}=\left( 16\times 100 \right)+36+480\]
\[\Rightarrow {{\left( 46 \right)}^{2}}=1600+480+36\]
\[\Rightarrow {{\left( 46 \right)}^{2}}=2116\]

Thus, the square of 46 is 2116.

Note: We can also solve this method by the following method: it is given in the question that we have to find the square of 46, i.e. \[{{\left( 46 \right)}^{2}}.\] The \[{{\left( 46 \right)}^{2}}\] will be obtained by multiplying 46 with 46, i.e. \[{{\left( 46 \right)}^{2}}=46\times 46.\]
To multiply 46 with 46, we will first multiply 6 with 6. Thus, we will get 36. We write 6 below the sixes and we will take 3 to the carry. Now, we will multiply 6 with 4. Thus, we will get 24. We will add the number in the carry, i.e. 3 to it. Thus, we will get 27. We will write 27 below both the 4s. Thus, we will have,
Step 1:
\[\begin{align}
  & \times \underline{\overset{3}{\mathop{\begin{align}
  & 46 \\
 & 46 \\
\end{align}}}\,} \\
 & ^{(24+3)6} \\
\end{align}\]
\[\begin{align}
  & \times \underline{\overset{3}{\mathop{\begin{align}
  & 46 \\
 & 46 \\
\end{align}}}\,} \\
 & \left( 276 \right) \\
\end{align}\]

Now, similarly, we will multiply 4 with 6 individually. When we will multiply 4 with 6, we will get 24. We will write 4 below 7 and put 0 below the 6. Now, we will take 2 to carry. Now, we will multiply 4 with 4 and add carry, i.e. 2 to it. Thus, we will get 16 + 2 = 18. We will write below 2. Thus, we will get,
Step 2:

\[\begin{align}
  & \times \underline{\overset{3}{\mathop{\begin{align}
  & 46 \\
 & 46 \\
\end{align}}}\,} \\
 & \text{ }276 \\
 & 1840 \\
\end{align}\]

Now, we will add the numbers obtained, i.e. 276 and 1840. Thus, we will get,
\[\begin{align}
  & \times \underline{\overset{3}{\mathop{\begin{align}
  & 46 \\
 & 46 \\
\end{align}}}\,} \\
 & \text{ }276 \\
 & \underline{+1840} \\
 & 2116 \\
\end{align}\]
Therefore, the square of $46$ is $2116$