Match exponents by balancing both sides: (9.95)^2 × (2.01)^3 = 2 × (?)^2. Find the nearest integer value of ? that satisfies the relation.

Introduction / Context:The expression equates a product of powers on the left to 2 times a square on the right. Solving for the base of the square requires isolating (?)^2 and taking a square root. Because the bases are near 10 and 2, the left-hand product is near 10^2 × 2^3 = 800, making the square root arithmetic tidy.

Given Data / Assumptions:

• LHS = (9.95)^2 × (2.01)^3
• RHS = 2 × (?)^2
• All numbers are positive; we want the positive root.

Concept / Approach:First compute the left-hand side approximately. Then set 2 * (?)^2 ≈ LHS, so (?) ≈ sqrt(LHS/2). Compare the result to the available integer options and select the nearest match.

Step-by-Step Solution:

Verification / Alternative check:Using tighter arithmetic gives ? ≈ 20.05. Among the options, 20 is the closest integer and balances both sides very well within the intended approximation level.

Why Other Options Are Wrong:

• 12, 15, 18: too small; squaring would yield much less than ~402.
• 25: too large; 25^2 = 625; 2*625 = 1250, well above the LHS.

Common Pitfalls:Taking square root before dividing by 2, or attempting to over-round bases (e.g., 9.95→9) which can skew the final square root noticeably.

Final Answer:20