Q10. - Algebra 1

Question

Find the area of rectangle.

F  $\mathbf{14}\sqrt{\mathbf{2}}$                     G 14                    H $\mathbf{98}\sqrt{\mathbf{2}}$                J $\mathbf{7}\sqrt{\mathbf{2}}$

Step-by-Step Solution

Answer

The area of the rectangle is $\mathbf{14}\sqrt{\mathbf{2}}$.

Option $\mathit{F}$ is correct.

Step 1. State the concept of radical symbol, radicand, index of radical and radical.

Radical symbol: The symbol $\sqrt{}$ square root or any $nth$ root symbol $\sqrt[n]{}$ (where n is a positive number) is called radical symbol.

Index of radical: The number above the root symbol in the radical expression is called the index of radical and it must be positive.

For example in the expression $\sqrt[4]{x}$, 4 is the index

Note: The index of $\sqrt{}$ is 2.

Radicand: The term inside the radical is called radicand.

Radical: The expression which contains radical symbol, index and radicand is called radical.

Step 2. State the rule of multiplication of radicals.

Two radicals can be only multiplied if the index of the terms are same. If the index of the terms are same, then the coefficient of the radicals are multiplied together and the radicands are multiplied together under a common radical symbol.

That is, $\left(a\sqrt{x}\right)\left(b\sqrt{y}\right)=(a\times b)\sqrt{x\times y}$

Step 3. State the concept of Area of rectangle.

Area of a rectangle is its length times width. Suppose the length of a rectangle is ‘$l$’ and its width is ‘$w$’ then the area of the rectangle ($A$) is given as:

$\text{Area}\left(A\right)=l\times w\dots \left(1\right)$

Step 4. Calculate the area.

Observe the figure given below.

From the figure, the length of the rectangle is $2\sqrt{14}$ and its width is $\sqrt{7}$

Substitute these values in $\left(1\right)$ to get the required area.

$\begin{array}{c}\text{Area}\left(A\right)=l\times w\\ =\left(2\sqrt{14}\right)\times \left(\sqrt{7}\right)\end{array}$

Here the index of both the terms are same, which is 2.

Therefore by multiplication rule,

  $\begin{array}{l}\text{Area}\left(A\right)=\left(2\sqrt{14}\right)\times \left(\sqrt{7}\right)\\ =\left(2\right)\left(\sqrt{14\times 7}\right)\\ =\left(2\right)\left(\sqrt{2\times 7\times 7}\right)\\ =\left(2\right)\left(\sqrt{2\times 7{}^2}\right)\\ =\left(2\right)\left(\sqrt{2}\right)\left(\sqrt{7^2}\right)\\ =2\left(\sqrt{2}\right)\left(7\right) \left[\sqrt{x^2}=x\right]\\ =(2\times 7)\left(\sqrt{2}\right)\\ =14\left(\sqrt{2}\right)\\ =14\sqrt{2}\end{array}$

Therefore the area of the given rectangle is $14\sqrt{2}$. Option $F$ is correct.