Question 1

Sum of two decimal square roots:  Evaluate √0.0009 + √0.01 and select the correct value.

Accepted Answer

Introduction / Context:Addition of square roots of small decimals requires careful decimal handling. Each root is easy, but the sum must be computed exactly and compared to the options. Given Data / Assumptions: Compute √0.0009 and √0.01 separately. Add the two results exactly. Concept / Approach:Use √(a/10^n) = √a / 10^(n/2) when n is even; here both decimals are perfect squares in decimal form. Step-by-Step Solution: √0.0009 = √(9/10^4) = 3/100 = 0.03.√0.01 = 0.1.Sum = 0.03 + 0.10 = 0.13. Verification / Alternative check:Square 0.03 to 0.0009 and 0.1 to 0.01 to confirm individual roots; then add. Why Other Options Are Wrong: 3 and 1/3 are orders of magnitude off. 0.3 is too large; 0.13 is not listed as a distinct selectable value except via “None of these.” Common Pitfalls:Confusing 0.0009 with 0.009 or misplacing decimal points when adding. Final Answer:None of these

Question 2

Add two simple square roots:  Compute √0.01 + √0.0064 and choose the correct sum.

Accepted Answer

Introduction / Context:The task is to add two exact decimal square roots. Each term is straightforward; accuracy depends on correct decimal placement. Given Data / Assumptions: √0.01 and √0.0064 are both perfect decimal roots. We will compute each and add. Concept / Approach:Recognize that 0.01 = 1/100 and 0.0064 = 64/10000. Then use square-root properties for fractions. Step-by-Step Solution: √0.01 = 0.1.√0.0064 = √(64/10000) = 8/100 = 0.08.Sum = 0.1 + 0.08 = 0.18. Verification / Alternative check:Square 0.08 to 0.0064 and 0.1 to 0.01; then add results as decimals. Why Other Options Are Wrong: 0.3 and 0.03 are incorrect totals. √0.18 is not equal to 0.18; that is a different number (~0.4243). The correct numeric value 0.18 is not explicitly listed, hence “None of these.” Common Pitfalls:Mistaking √0.18 for 0.18 or slipping a decimal place in 0.0064. Final Answer:None of these

Question 3

Approximate the square root of a non-perfect decimal:  What is a good decimal approximation for √0.9?

Accepted Answer

Introduction / Context:Some square roots are irrational and must be approximated. √0.9 is slightly less than 1, and a reasonable decimal approximation is required among the choices provided. Given Data / Assumptions: We compare the options to the true value of √0.9. 0.9 is near 1; the root should be near 1 but smaller. Concept / Approach:Use the idea that √(1 − ε) ≈ 1 − ε/2 for small ε. Here ε = 0.1, giving an estimate near 0.95. We then choose the closest option. Step-by-Step Solution: Approximation: √0.9 ≈ 1 − 0.1/2 = 0.95.More precisely, √0.9 ≈ 0.948683…Among the options, 0.94 is the closest sensible approximation. Verification / Alternative check:Square 0.94: 0.8836 (a bit low). Square 0.95: 0.9025 (a bit high). The true value is between; 0.94 is the nearest available option. Why Other Options Are Wrong: 0.3, 0.03, 0.33 are far too small. 0.9 is the radicand, not its square root. Common Pitfalls:Guessing 0.9 or 1; remember square roots compress values between 0 and 1 upward but remain below 1. Final Answer:0.94

Question 4

Simplify a radical expression by factoring:  Evaluate (√24 + √216) / √96.

Accepted Answer

Introduction / Context:This expression simplifies elegantly by factoring out common square-free parts from each square root. Recognizing multiples of 6 under the radicals is the key. Given Data / Assumptions: Expression: (√24 + √216) / √96. All radicals are positive principal roots. Concept / Approach:Rewrite each radicand as a product of a perfect square and a square-free part, then cancel common factors between numerator and denominator. Step-by-Step Solution: √24 = √(4 * 6) = 2√6.√216 = √(36 * 6) = 6√6.Numerator = 2√6 + 6√6 = 8√6.√96 = √(16 * 6) = 4√6.Therefore, (√24 + √216) / √96 = (8√6) / (4√6) = 2. Verification / Alternative check:Numeric check: √24 ≈ 4.899, √216 ≈ 14.697, sum ≈ 19.596; √96 ≈ 9.798; ratio ≈ 2.00. Why Other Options Are Wrong: 2√6, 6√2, 2/√6, √6 do not equal the simplified value of the ratio. Common Pitfalls:Forgetting to factor out the perfect squares or failing to cancel √6 correctly. Final Answer:2

Question 5

Rationalize a reciprocal of radicals:  Compute 1 / (√9 − √8) and express it in simplest surd form.

Accepted Answer

Introduction / Context:Rationalizing denominators with surds is a classic skill. Using conjugates eliminates radicals in the denominator and yields a clean exact expression. Given Data / Assumptions: Expression: 1 / (√9 − √8) = 1 / (3 − 2√2). We use the conjugate (3 + 2√2). Concept / Approach:Multiply numerator and denominator by the conjugate to create a difference of squares in the denominator. Step-by-Step Solution: 1 / (3 − 2√2) * (3 + 2√2)/(3 + 2√2) = (3 + 2√2) / (9 − (2√2)^2).(2√2)^2 = 4 * 2 = 8.Denominator = 9 − 8 = 1.Therefore, the expression simplifies to 3 + 2√2. Verification / Alternative check:Numerical check: 3 + 2√2 ≈ 3 + 2*1.414 = 5.828; 1/(3 − 2*1.414) ≈ 1/(0.172) ≈ 5.814 (difference due to rounding of √2); exact values match symbolically. Why Other Options Are Wrong: (3 − 2√2)/2 and (3 − 2√2) are not equal to the simplified reciprocal. 1/(3 + 2√2) is the reciprocal of the correct answer. Common Pitfalls:Forgetting that (a − b)(a + b) = a^2 − b^2 and mis-squaring 2√2. Final Answer:(3 + 2√2)

Question 6

Clarify grouping and evaluate:  Compute √(0.16 / 0.4) and choose the correct decimal approximation.

Accepted Answer

Introduction / Context:The original text can be ambiguous; consistent with the options, interpret it as √(0.16 / 0.4). We simplify the fraction first, then take the square root and compare to the listed choices. Given Data / Assumptions: Evaluate √(0.16 / 0.4). Use principal (positive) square roots. Concept / Approach:Simplify the ratio inside the root: 0.16 / 0.4 = 16/40 = 2/5 = 0.4. Then take √0.4 and express as a decimal or simple surd if needed. Step-by-Step Solution: 0.16 / 0.4 = 0.4.√0.4 = √(4/10) = √(2/5) = (√10)/5 ≈ 0.632455...Rounded to two decimals, this is about 0.63. Verification / Alternative check:Square 0.63 ≈ 0.3969, close to 0.4; using exact (√10)/5 squared is exactly 0.4. Why Other Options Are Wrong: 0.2 and 0.02 are too small. 2√5/5 ≈ 0.894 is too large. 1 is the value of √0.16/0.4 if mis-grouped; with correct grouping the value is ≈ 0.632. Common Pitfalls:Ignoring parentheses and computing (√0.16)/0.4 = 1; the intended grouped form avoids this ambiguity. Final Answer:0.63

Question 7

Square root of a small decimal:  Find an appropriate approximation for √0.064.

Accepted Answer

Introduction / Context:We estimate the square root of a small decimal. Because 0.064 is not a perfect square, we present a rounded value consistent with the options provided. Given Data / Assumptions: Number: 0.064 = 64/1000. We want √0.064. Concept / Approach:Compute √(64/1000) = 8/√1000 = 8/(10√10) = (4)/(5√10). Convert to decimal using √10 ≈ 3.1623. Step-by-Step Solution: √0.064 = (4)/(5√10) ≈ 4 / (5 * 3.1623) ≈ 4 / 15.8115 ≈ 0.25298.Rounded to three decimals, ≈ 0.253; the nearest listed option is 0.252. Verification / Alternative check:Square 0.252 ≈ 0.063504, very close to 0.064; with 0.253^2 ≈ 0.064009, straddling the true value. Why Other Options Are Wrong: 0.8, 0.08, 0.008 are order-of-magnitude mismatches. 0.32 is also too large. Common Pitfalls:Assuming 0.064 is close to 0.06 and guessing; compute using √(64/1000) to stay accurate. Final Answer:0.252

Question 8

Square root recognition:  Evaluate √0.121 and choose the closest value.

Accepted Answer

Introduction / Context:We compute the square root of a decimal 0.121 = 121/1000. Unlike 0.0121 (which is 0.11^2), this value is larger and needs an appropriate approximation. Given Data / Assumptions: Number: 0.121 = 121/1000. We want √0.121. Concept / Approach:Use a decimal approximation: √0.121 is between √0.09 (= 0.3) and √0.16 (= 0.4). Closer to 0.11? No—note 0.11^2 = 0.0121, not 0.121. Step-by-Step Solution: 0.121 = 121/1000 ≈ 0.121.Try 0.347: 0.347^2 = 0.120409, very close.Thus √0.121 ≈ 0.348; among choices 0.347 is closest. Verification / Alternative check:More accurate value: √0.121 ≈ 0.348552; option 0.347 differs by about 0.0016, within typical rounding. Why Other Options Are Wrong: 0.11 corresponds to √0.0121, not √0.121. 1.1 and 0.011 are off by factors of 10 or 100. 0.321 is not close enough. Common Pitfalls:Confusing 0.121 with 0.0121; always square the guess to verify. Final Answer:0.347

Question 9

Cube root of a very small decimal:  Find the cube root of 0.000027.

Accepted Answer

Introduction / Context:We need the cube root of a small decimal. Recognize familiar cubes: 27 is 3^3, so 0.000027 relates directly to a scaled cube. Given Data / Assumptions: Number: 0.000027 = 27 / 10^6. Use principal cube root (positive). Concept / Approach:For positive a, ∛(a/10^n) = ∛a / 10^(n/3) when n is a multiple of 3. Here n = 6 is divisible by 3, which makes evaluation straightforward. Step-by-Step Solution: 0.000027 = 27 / 10^6.∛(27 / 10^6) = ∛27 / ∛(10^6) = 3 / 10^2 = 3/100 = 0.03.Therefore, cube root = 0.03. Verification / Alternative check:(0.03)^3 = 27 / 10^6 = 0.000027, confirming the result exactly. Why Other Options Are Wrong: .3^3 = 0.027, too large by a factor of 1000. .003^3 = 2.7e-8, far too small. “None of these” is not needed since .03 is present. Common Pitfalls:Mixing square-root logic with cube roots or miscounting decimal-place triplets for cube roots. Final Answer:.03

Question 10

Find the least positive integer which, when multiplied with 74088, makes the product a perfect square. Show the reasoning clearly and retain the numerical value 74088 without approximation.

Accepted Answer

Introduction / Context: This question tests prime factorization and the condition for a number to be a perfect square. A product is a perfect square if every prime factor appears with an even exponent in its prime factorization. Given Data / Assumptions: Number given: 74088. We seek the smallest positive integer k such that 74088 * k is a perfect square. No approximation is needed; work with exact prime powers. Concept / Approach: Factorize 74088 into primes. For a perfect square, every prime's exponent must be even. If an exponent is odd, multiply by that prime once to make it even. Combine all such missing primes for the least multiplier. Step-by-Step Solution: 74088 = 2^3 * 3^3 * 7^3 (by prime factorization). Each exponent (3) is odd for 2, 3, and 7. To make exponents even, multiply by one 2, one 3, and one 7. Least multiplier k = 2 * 3 * 7 = 42. Now 74088 * 42 = 2^(3+1) * 3^(3+1) * 7^(3+1) = 2^4 * 3^4 * 7^4, a perfect square. Verification / Alternative check: A quick check: any odd exponent becomes even by multiplying once with that prime. All three primes required exactly one copy, confirming k = 42 is minimal. Why Other Options Are Wrong: 44 = 2^2 * 11 introduces 11 and leaves 3 and 7 odd. 46 = 2 * 23 introduces 23 and leaves 3 and 7 odd. 48 = 2^4 * 3 leaves 7 odd; also adds unnecessary powers. Common Pitfalls: Students sometimes add large multipliers or multiply by existing odd exponents directly without ensuring minimality. Always fix each odd exponent by multiplying once by that prime only. Final Answer: 42

Question 11

Compute the exact value of √248 + √(52 + √144). Express the final result numerically (rounded to two decimals if needed) and clearly show your steps.

Accepted Answer

Introduction / Context: This arithmetic problem involves nested square roots and evaluates to a real number. It tests the ability to simplify square roots step by step and handle nesting correctly. Given Data / Assumptions: Expression: √248 + √(52 + √144). Use exact simplification where possible (e.g., √144 = 12). Round to two decimals if the result is irrational. Concept / Approach: Simplify inner radicals first, then compute the outer roots. Recognize perfect squares to keep the computation clean. Combine results at the end. Step-by-Step Solution: Compute √144 = 12. Then 52 + √144 = 52 + 12 = 64. So √(52 + √144) = √64 = 8. Next evaluate √248. Note 248 = 4 * 62, so √248 = √4 * √62 = 2√62 ≈ 2 * 7.8740 = 15.748. Finally, √248 + √(52 + √144) ≈ 15.748 + 8 = 23.748. Rounded to two decimals: 23.75. Verification / Alternative check: A calculator check: √62 ≈ 7.8740; doubling gives 15.748. Add 8 to get 23.748, matching our rounded 23.75. Why Other Options Are Wrong: 22.00 and 20.00 are too low; they ignore the exact contribution from √248. 24.00 is close but slightly high; the exact computation yields approximately 23.75, not 24. Common Pitfalls: Misreading √(52 + √144) as √52 + √144 or approximating √248 as √256 (16) leads to incorrect answers. Always simplify the nested radical correctly before summing. Final Answer: 23.75

Question 12

Evaluate √176 + √2401. Provide the exact simplified value where possible and give the final sum (rounded to two decimals if needed).

Accepted Answer

Introduction / Context: This computation checks simplification of square roots and addition of an irrational and a perfect square root. It is a direct arithmetic task with careful simplification. Given Data / Assumptions: Expression: √176 + √2401. 2401 is a perfect square (49^2). Round the final decimal to two places as necessary. Concept / Approach: Simplify each root separately. For perfect squares, take exact values. For non-perfect squares, factor out perfect square parts if possible, or evaluate numerically. Step-by-Step Solution: √2401 = 49 because 49^2 = 2401. For √176: factor 176 = 16 * 11, so √176 = √16 * √11 = 4√11. Approximate √11 ≈ 3.3166, thus √176 ≈ 4 * 3.3166 = 13.2664. Sum: √176 + √2401 ≈ 13.2664 + 49 = 62.2664. Rounded to two decimals: 62.27. Verification / Alternative check: A quick calculator check of √11 confirms ≈ 3.3166; multiplying by 4 and adding 49 reproduces ≈ 62.27. Why Other Options Are Wrong: 64.00 and 60.00 are rounded guesses not supported by precise simplification. 56.50 is much too small; √2401 alone contributes 49. Common Pitfalls: Treating √176 as √169 (13) or √196 (14) is an error. Always factor to extract perfect squares or compute accurately. Final Answer: 62.27

Question 13

Simplify the expression (√32 + √48) / (√8 + √12), and provide the exact simplified value.

Accepted Answer

Introduction / Context: This problem tests simplification of expressions with square roots using factorization of perfect squares and cancellation. The goal is an exact simplified value without decimals. Given Data / Assumptions: Expression: (√32 + √48) / (√8 + √12). All terms are positive real numbers. Concept / Approach: Express each radicand as a product including a perfect square to simplify: √32 = √(16*2) = 4√2, √48 = √(16*3) = 4√3, √8 = √(4*2) = 2√2, √12 = √(4*3) = 2√3. Then factor and cancel common terms. Step-by-Step Solution: √32 = 4√2 and √48 = 4√3. √8 = 2√2 and √12 = 2√3. Numerator: 4√2 + 4√3 = 4(√2 + √3). Denominator: 2√2 + 2√3 = 2(√2 + √3). Ratio = [4(√2 + √3)] / [2(√2 + √3)] = 4 / 2 = 2. Verification / Alternative check: Substitute √2 ≈ 1.414 and √3 ≈ 1.732 to compute both numerator and denominator and confirm the quotient is 2.0. Why Other Options Are Wrong: √2, 4, and 8 do not follow from the exact cancellations; only 2 results after factoring common (√2 + √3). Common Pitfalls: Forgetting to factor out common radicals or attempting to rationalize unnecessarily can complicate the problem. Always reduce radicals first. Final Answer: 2

Question 14

Given √3 ≈ 1.732 and √2 ≈ 1.414, compute the value of 1 / (√3 + √2). Provide the value to three decimal places.

Accepted Answer

Introduction / Context: This numerical approximation problem practices handling radicals and reciprocals. Rationalization gives an exact form, while given approximations yield the decimal result. Given Data / Assumptions: √3 ≈ 1.732, √2 ≈ 1.414. Expression: 1 / (√3 + √2). Round to three decimal places. Concept / Approach: Using the conjugate: 1 / (√3 + √2) = (√3 − √2) / [(√3 + √2)(√3 − √2)] = (√3 − √2) / (3 − 2) = √3 − √2. Then substitute the approximations. Step-by-Step Solution: 1 / (√3 + √2) = √3 − √2. ≈ 1.732 − 1.414 = 0.318. To three decimals, the value is 0.318. Verification / Alternative check: Direct division: 1 / (1.732 + 1.414) ≈ 1 / 3.146 ≈ 0.3179, which rounds to 0.318 and matches the conjugate method. Why Other Options Are Wrong: 0.064 and 0.308 deviate from the rationalized exact form √3 − √2. 2.146 is vastly too large for a reciprocal of a number greater than 3. Common Pitfalls: Forgetting to rationalize or incorrectly subtracting decimal approximations may produce small but significant errors. Always check with both methods if unsure. Final Answer: 0.318

Question 15

Using √6 ≈ 2.55 (and standard approximations √2 and √3), evaluate √(2/3) + 3√(3/2). Round to two decimal places.

Accepted Answer

Introduction / Context: This question checks comfort with radicals and fractional arguments. It involves combining √(2/3) and 3√(3/2) accurately to reach a numerical result. Given Data / Assumptions: Approximation: √6 ≈ 2.55 (useful via √(2/3) = √2/√3 and √(3/2) = √3/√2). We can use √2 ≈ 1.414 and √3 ≈ 1.732. Expression: √(2/3) + 3√(3/2). Concept / Approach: Convert each term using separate square roots: √(2/3) = √2 / √3 and √(3/2) = √3 / √2. Evaluate numerically and sum, respecting multiplication by 3 for the second term. Step-by-Step Solution: √(2/3) = √2 / √3 ≈ 1.414 / 1.732 ≈ 0.8165. √(3/2) = √3 / √2 ≈ 1.732 / 1.414 ≈ 1.2247. 3√(3/2) ≈ 3 * 1.2247 = 3.6741. Total ≈ 0.8165 + 3.6741 = 4.4906. Rounded to two decimals: 4.49. Verification / Alternative check: Cross-check using √6 ≈ 2.55 is consistent since (√3/√2)*(√2/√3) = 1; the numerical evaluations agree with standard approximations. Why Other Options Are Wrong: 4.48 and 4.50 are near but not as accurate as 4.49 with the given approximations. “None of these” is incorrect because we have a clear matching value. Common Pitfalls: Mixing up √(a/b) with √a/√b and forgetting the factor 3 in the second term lead to common errors. Keep track of multipliers. Final Answer: 4.49

Question 16

If (√2)^n = 64, determine the value of n. (Interpretation: the expression denotes (√2) raised to the power n equals 64.)

Accepted Answer

Introduction / Context: Exponent rules are central in this problem. Recognizing powers of 2 and rewriting square roots in exponential form lead to a quick solution. Given Data / Assumptions: (√2)^n = 64. 64 = 2^6. √2 = 2^(1/2). Concept / Approach: Convert √2 to exponential form, then equate exponents for equal bases. If 2^(n/2) = 2^6, then n/2 = 6 and n follows immediately. Step-by-Step Solution: √2 = 2^(1/2). (√2)^n = (2^(1/2))^n = 2^(n/2). Given 2^(n/2) = 64 = 2^6. Therefore n/2 = 6 ⇒ n = 12. Verification / Alternative check: Substitute back: (√2)^12 = (2^(1/2))^12 = 2^6 = 64. Correct. Why Other Options Are Wrong: 2, 4, 6 do not satisfy 2^(n/2) = 2^6. Common Pitfalls: Misreading √2^n as √(2^n) rather than (√2)^n changes the equation. The wording here clarifies it should be interpreted as (√2)^n. Final Answer: 12

Question 17

Evaluate the expression √(1.21 × 0.9) ÷ √(1.1 × 0.11) and provide the exact simplified value.

Accepted Answer

Introduction / Context: This tests manipulating radicals and recognizing that √a / √b = √(a/b). Arithmetic with decimals is straightforward when converted to fractions or simplified decimals. Given Data / Assumptions: Compute √(1.21 × 0.9) / √(1.1 × 0.11). Concept / Approach: Combine the radicals: √(1.21 × 0.9) / √(1.1 × 0.11) = √[(1.21 × 0.9) / (1.1 × 0.11)]. Then simplify the quotient inside the square root. Step-by-Step Solution: Compute numerators/denominators: 1.21 × 0.9 = 1.089. 1.1 × 0.11 = 0.121. Form the ratio: 1.089 / 0.121 = 9 (since 0.121 × 9 = 1.089). Thus the expression is √9 = 3. Verification / Alternative check: Write in fractions: 1.21 = 121/100, 0.9 = 9/10, 1.1 = 11/10, 0.11 = 11/100. The ratio becomes √[(121/100 * 9/10) / (11/10 * 11/100)] = √[ (121*9*100) / (100*10*121) ] = √9 = 3. Why Other Options Are Wrong: 2, 9, 11 do not result from the exact simplification of the given decimals. Common Pitfalls: Handling decimals without grouping under one radical often leads to arithmetic slips. Combine under a single square root to simplify cleanly. Final Answer: 3

Question 18

If √(1 + 27/169) = 1 + N/13, determine the integer N that satisfies the identity.

Accepted Answer

Introduction / Context: This problem checks algebraic manipulation and recognition of perfect squares under radicals. Converting to a single fraction inside the square root simplifies the comparison. Given Data / Assumptions: Equation: √(1 + 27/169) = 1 + N/13. N is an integer. Concept / Approach: Evaluate the left-hand side exactly by combining fractions. Compare the simplified exact value with the right-hand side to find N. Step-by-Step Solution: Compute inside radical: 1 + 27/169 = (169/169) + (27/169) = 196/169. √(196/169) = √196 / √169 = 14 / 13. Set 14/13 = 1 + N/13. Then N/13 = 14/13 − 1 = 1/13 ⇒ N = 1. Verification / Alternative check: Substitute N = 1 into the RHS: 1 + 1/13 = 14/13, which matches the LHS exactly. Why Other Options Are Wrong: 3, 5, and 7 give 1 + N/13 values greater than 14/13 and thus do not equal √(196/169). Common Pitfalls: Forgetting that √(a/b) = √a / √b, or approximating decimals instead of using exact fractions can create rounding errors. Here, exact arithmetic is clean and quick. Final Answer: 1

Question 19

Evaluate the infinite nested radical x = √(12 + √(12 + √(12 + …))). Find the exact value of x.

Accepted Answer

Introduction / Context: Infinite nested radicals can often be evaluated by setting the entire expression equal to a variable and exploiting self-similarity. This tests algebraic reasoning and solving a quadratic equation arising from the structure. Given Data / Assumptions: x = √(12 + √(12 + √(12 + …))). Assume convergence to a finite positive x (standard for such forms with positive constants). Concept / Approach: Because the structure repeats, set x = √(12 + x). Then square both sides and rearrange to a quadratic in x. Solve for positive x (as square roots yield nonnegative values). Step-by-Step Solution: Let x = √(12 + x). Square: x^2 = 12 + x. Rearrange: x^2 − x − 12 = 0. Solve quadratic: x = [1 ± √(1 + 48)] / 2 = [1 ± 7] / 2. Possible roots: x = 4 or x = −3. Since x is a value of a square root expression, x ≥ 0 ⇒ x = 4. Verification / Alternative check: Substitute x = 4 back: √(12 + 4) = √16 = 4, which satisfies the self-referential equation. Why Other Options Are Wrong: 3 and 6 do not satisfy x = √(12 + x). “Greater than 6” contradicts the exact solution x = 4. Common Pitfalls: Including the negative root from the quadratic or attempting to expand the radical indefinitely rather than using self-similarity. Final Answer: 4

Question 20

Given √2 ≈ 1.4142, compute the value of 7 / (3 + √2). Provide the result to four decimal places.

Accepted Answer

Introduction / Context: This is a straightforward evaluation of a rational expression involving a radical in the denominator. Rationalization provides insight; approximation provides a quick decimal answer. Given Data / Assumptions: √2 ≈ 1.4142. Expression: 7 / (3 + √2). Round to four decimal places. Concept / Approach: Either directly substitute √2 and compute, or rationalize by multiplying numerator and denominator by (3 − √2) to check the magnitude. Both lead to the same decimal result. Step-by-Step Solution: Direct evaluation: denominator = 3 + 1.4142 = 4.4142. Value = 7 / 4.4142 ≈ 1.5858. Verification / Alternative check: Rationalize: 7 / (3 + √2) = 7(3 − √2) / (9 − 2) = 7(3 − √2) / 7 = 3 − √2 ≈ 3 − 1.4142 = 1.5858, confirming the result. Why Other Options Are Wrong: 4.4142 is the denominator, not the value. 3.4852 and 3.5858 are unrelated approximations and not equal to 3 − √2. Common Pitfalls: Confusing the denominator with the final result, or forgetting to round consistently. Final Answer: 1.5858

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