Question

8. Darrin was asked to simplify √72 and arrived at the solution 2√18. Is this solution completely simplified? Explain why or why not. Include your work to justify your response.

Answer 1

No

$6\sqrt[]{2}$

Explanation

Step 1

let the given expression

$\sqrt[]{72}$

the firs thing to do is to get the prime factors of 72, so

a) prime factors

$\begin{gathered} 72=2\cdot2\cdot2\cdot3\cdot3 \\ \text{hence} \\ 72=2^33^2 \\ 72=2\cdot2^2\cdot3^2 \end{gathered}$

Step 2

now we know that:

$\begin{gathered} \sqrt[n]{a^n}\text{ =a} \\ \text{and} \\ \sqrt[]{ab}=\sqrt[]{a}\cdot\sqrt[]{b} \end{gathered}$

so

$\begin{gathered} \sqrt[]{72}=\sqrt[]{2\cdot2^2\cdot3^2} \\ \sqrt[]{72}=\sqrt[]{2}\sqrt[]{2^2}\sqrt[]{3^2} \\ \sqrt[]{72}=2\cdot3\sqrt[]{2} \\ \sqrt[]{72}=6\sqrt[]{2} \end{gathered}$

therefotre:

Darrin was wrong because his expression was not totally simplified, the full simplification is

$6\sqrt[]{2}$

I hope this helps you