Question

question 3.
melanie is looking at her notes on simplifying radicals. she forgot to write the complete problem and knows that the radical expression simplified begins with $6\sqrt{}$.
which of the following radical expressions can be simplified to an expression that begins with $6\sqrt{}$?
a. $\sqrt{78}$
b. $\sqrt{216}$
c. $\sqrt{66}$
d. $\sqrt{156}$

Explanation:

Step1: Reverse the simplification process

If the simplified form is $6\sqrt{x}$, then the original radical is $\sqrt{6^2 \times x} = \sqrt{36x}$. We need to find which option can be written as $\sqrt{36x}$ where $x$ is an integer with no perfect square factors other than 1.

Step2: Test Option A

$\sqrt{78}$: 78 divided by 36 is $\frac{78}{36} = \frac{13}{6}$, not an integer. So this cannot be written as $\sqrt{36x}$.

Step3: Test Option B

$\sqrt{216}$: Calculate $216 \div 36 = 6$. So $\sqrt{216} = \sqrt{36 \times 6} = 6\sqrt{6}$, which matches the form $6\sqrt{x}$.

Step4: Verify remaining options (optional)

$\sqrt{66} \div 36 = \frac{11}{6}$, not an integer. $\sqrt{156} \div 36 = \frac{13}{3}$, not an integer.

Answer:

B. $\sqrt{216}$