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.

Answer 2

This question gives the equation \( \frac{?}{\sqrt{128}} = \frac{\sqrt{162}}{?} \) and asks us to find the missing number. Since the same unknown appears on both sides, one direct way to check is to substitute each option back into the equation and see which one keeps both sides equal.

1. 12: Substituting gives \( \frac{12}{\sqrt{128}} \) on the left and \( \frac{\sqrt{162}}{12} \) on the right. Since \( \sqrt{128} \approx 11.31 \) and \( \sqrt{162} \approx 12.73 \), the left side is about \( \frac{12}{11.31} \approx 1.06 \) and the right side is about \( \frac{12.73}{12} \approx 1.06 \). Both sides match, so this value satisfies the equation.
2. 14: The left side becomes \( \frac{14}{11.31} \approx 1.24 \), while the right side becomes \( \frac{12.73}{14} \approx 0.91 \). These are not equal, so 14 does not work.
3. 144: The left side becomes \( \frac{144}{11.31} \approx 12.73 \), while the right side becomes \( \frac{12.73}{144} \approx 0.09 \). These are far apart, so 144 does not satisfy the equation.
4. 196: The left side becomes \( \frac{196}{11.31} \approx 17.33 \), while the right side becomes \( \frac{12.73}{196} \approx 0.06 \). These do not match either, so 196 is not correct.

Only substituting 12 keeps both sides of the equation equal, confirming it as the value that makes the relation true.

Therefore, the correct answer is 12.