Question

Find ? in equation: ?/\(\sqrt{128}\) = \(\sqrt{162}\)/?

• A. 12
• B. 14
• C. 144
• D. 196

Hint:

Always try to simplify square roots by factoring out perfect squares before performing multiplication. This often makes the calculation much easier and reduces the chance of errors. For example, $\sqrt{128} = \sqrt{64 \times 2}$ rather than directly multiplying large numbers.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem asks to find the missing value (represented by '?') in a given algebraic equation involving square roots.

Step 2: Key Formula or Approach:
1. Rearrange the equation to isolate the unknown.
2. Simplify square roots where possible ($ \sqrt{ab} = \sqrt{a}\sqrt{b} $).
3. Solve for the unknown.

Step 3: Detailed Explanation:
Given equation:
\[ \frac{?}{\sqrt{128}} = \frac{\sqrt{162}}{?} \]
Let the unknown value be \( x \).
\[ \frac{x}{\sqrt{128}} = \frac{\sqrt{162}}{x} \]
Cross-multiply:
\[ x^2 = \sqrt{128} \times \sqrt{162} \]
Simplify the square roots:
\[ \sqrt{128} = \sqrt{64 \times 2} = 8\sqrt{2} \]
\[ \sqrt{162} = \sqrt{81 \times 2} = 9\sqrt{2} \]
Substitute these simplified values back into the equation for \( x^2 \):
\[ x^2 = (8\sqrt{2}) \times (9\sqrt{2}) \]
\[ x^2 = 8 \times 9 \times \sqrt{2} \times \sqrt{2} \]
\[ x^2 = 72 \times 2 \]
\[ x^2 = 144 \]
Solve for \( x \):
\[ x = \sqrt{144} = 12 \]

Step 4: Final Answer:
The missing value is 12.