Question

Solve seven square root three plus two square root nine and explain whether the answer is rational or irrational.

Answer 1

The answer to the given question is irrational.

Explanation:

A rational number can be expressed as a fraction, while irrational number is a real number that cannot be expressed as a simple fraction.

From the question,

`7√(3) + 2√(9)` =
`7√(3) + 2\sqrt{3^(2) }`

=
`7√(3) + 2(3)^{(2)/(2) }`

=
`7√(3) + 6`

But,
`7√(3) + 6` would give an irrational answer.

Thus, seven square root three plus two square root nine equals 18.12435565 which is irrational.

Answer 2

Seven square root three plus two square root nine is irrational and the value is 18.12

SOLUTION:

Given, seven square root three plus two square root nine

= seven square root three + two square root nine

`\begin{array}{l}{=7 \sqrt[2]{3}+2 \sqrt[2]{9}} \\ {=7 \sqrt[2]{3}+2 \sqrt[2]{3^(2)}} \\ {=7 \sqrt[2]{3}+2 * 3} \\ {=7 \sqrt[2]{3}+6}\end{array}`

We know that,
`\sqrt[2]{3}` is irrational as it cannot be expressed as ratio of two integers.

Now,
`7 \sqrt[2]{3}+6` is also irrational, because sum of rational and irrational is always irrational.

`\begin{array}{l}{\text { On solving } 7 \sqrt[2]{3}+6,} \\ {7 \sqrt[2]{3}+6=7(1.732)+6=18.12}\end{array}`

Hence, seven square root three plus two square root nine is irrational.