Question

Compare the surds: $7\sqrt{2}$ and $5\sqrt{3}$ .

Answer 1

Step 1. Square the numbers.

The values inside the square root which cannot be further subdivided into integers are called surds.

Here, the surds are given to compare are $7\sqrt{2}$ and $5\sqrt{3}$

Square the surds:

$$\begin{gathered} {\left(7\sqrt{2}\right)}^2=7\sqrt{2}\times 7\sqrt{2} \\ \Rightarrow {\left(7\sqrt{2}\right)}^2=7\times \sqrt{2}\times 2\times \sqrt{2} \\ \Rightarrow {\left(7\sqrt{2}\right)}^2=49\times 2 \\ \Rightarrow {\left(7\sqrt{2}\right)}^2=98 \end{gathered}$$

And

$$\begin{gathered} {\left(5\sqrt{3}\right)}^2=5\sqrt{3}\times 5\sqrt{3} \\ \Rightarrow {\left(5\sqrt{3}\right)}^2=5\times \sqrt{3}\times 5\times \sqrt{3} \\ \Rightarrow {\left(5\sqrt{3}\right)}^2=25\times 3 \\ \Rightarrow {\left(5\sqrt{3}\right)}^2=75 \end{gathered}$$

Step 2. Compare the numbers.

Compare the square of $7\sqrt{2}$ and $5\sqrt{3}$ which are $98$ and $75$ respectively:

$$\begin{gathered} 98>75 \\ \Rightarrow 7\sqrt{2}>5\sqrt{3} \end{gathered}$$

Hence, the $7\sqrt{2}$ is greater than $5\sqrt{3}$ .