Question

What is the fourth proportional to $\sqrt{6}$, $\sqrt{8}$ and $\sqrt{21}$ ?

• A. $3\sqrt{7}$
• B. $2\sqrt{7}$
• C. $8\sqrt{7}$
• D. $5\sqrt{7}$

Hint:

When dealing with square roots in ratios, group them under a single radical sign before simplifying.
This reduces computation errors and makes the fraction easy to divide.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The goal of this question is to find the fourth proportional to three given mathematical surd values: $\sqrt{6}$, $\sqrt{8}$, and $\sqrt{21}$.

Step 2: Key Formula or Approach:
If $a$, $b$, and $c$ are three quantities, then their fourth proportional $x$ satisfies the ratio:
\[ \frac{a}{b} = \frac{c}{x} \implies x = \frac{b \times c}{a} \]

Step 3: Detailed Explanation:
Let us substitute the given values into the formula to calculate $x$:

• Let $a = \sqrt{6}$, $b = \sqrt{8}$, and $c = \sqrt{21}$.

• Set up the proportion equation:
\[ x = \frac{\sqrt{8} \times \sqrt{21}}{\sqrt{6}} \]

• Combine the terms under a single square root radical using surd properties:
\[ x = \sqrt{\frac{8 \times 21}{6}} \]

• Simplify the fraction inside the square root step-by-step:
\[ \frac{8 \times 21}{6} = \frac{168}{6} = 28 \]

• Simplify the resulting square root:
\[ x = \sqrt{28} = \sqrt{4 \times 7} = 2\sqrt{7} \]

Step 4: Final Answer:
The fourth proportional to the three values is $2\sqrt{7}$, which matches option (B).