Question 1

Rationalize and simplify the surd expression: If (5 + 2√3) / (7 + 4√3) = a + b√3, determine the exact values of a and b.

Accepted Answer

Introduction / Context: Converting a fraction with surds into the form a + b√3 requires rationalizing the denominator. By multiplying numerator and denominator by the conjugate of the denominator, we eliminate the surd from the denominator and then collect rational and irrational parts separately. Given Data / Assumptions: Original expression: (5 + 2√3) / (7 + 4√3). Target form: a + b√3 where a and b are rational numbers. Conjugate of (7 + 4√3) is (7 − 4√3). Concept / Approach: Multiply numerator and denominator by the conjugate (7 − 4√3). Use (p + q√3)(p − q√3) = p^2 − 3q^2 to remove the surd from the denominator. Then expand the numerator and separate terms with and without √3 to identify a and b. Step-by-Step Solution: Compute denominator: (7 + 4√3)(7 − 4√3) = 49 − 3·(4^2) = 49 − 48 = 1.Compute numerator: (5 + 2√3)(7 − 4√3) = 35 − 20√3 + 14√3 − 8·3.Simplify numerator: 35 − 6√3 − 24 = 11 − 6√3.Thus, (5 + 2√3)/(7 + 4√3) = 11 − 6√3 ⇒ a = 11, b = −6. Verification / Alternative check: Because the rationalized denominator equals 1, the product in the numerator is already the final simplified value. Quick numerical check: evaluate both sides approximately to confirm consistency. Why Other Options Are Wrong: The signs of a and b are determined by the cross terms (−20√3 + 14√3 = −6√3). Any option with b positive or wrong a-value ignores this cancellation. Common Pitfalls: Forgetting that (√3)^2 = 3 when expanding, or using the wrong conjugate sign leading to an incorrect denominator. Final Answer: a = 11, b = -6

Question 2

Evaluate the power tower with surds: Compute the exact value of [(√2)^(√2)]^(√2). Choose the best classification of the result.

Accepted Answer

Introduction / Context: Exponent laws remain valid for positive real bases and real exponents: (a^b)^c = a^(b*c). This problem applies that idea to a surd base √2 and two surd exponents √2. The goal is to simplify the nested exponents and then classify the final value in the standard number hierarchy. Given Data / Assumptions: Expression: [(√2)^(√2)]^(√2). Base a = √2 > 0, so exponent laws for real exponents apply. All operations are real-valued. Concept / Approach: Use the law (a^b)^c = a^(b*c). Multiplying the two √2 exponents gives √2 * √2 = 2. Thus the entire tower collapses to (√2)^2, which is an exact integer. We then identify its set membership: integer, natural, rational, real. Step-by-Step Solution: Start with E = [(√2)^(√2)]^(√2).Apply exponent law: E = (√2)^(√2 * √2) = (√2)^(2).Evaluate: (√2)^2 = 2. Verification / Alternative check: Compute numerically: √2 ≈ 1.4142. Then (√2)^(√2) ≈ 1.4142^1.4142 ≈ 1.632..., and raising that to √2 returns exactly 2 by the algebraic identity used. Why Other Options Are Wrong: The value 2 is an integer and a natural number (in the usual positive-integer convention). Therefore it is not “non-integer rational” nor “irrational.” Common Pitfalls: Believing that a surd base automatically produces an irrational result after exponentiation. Here, the structure of exponents forces an exact integer after simplification. Final Answer: a natural number

Question 3

Simplify the compound monomial quotient: Simplify (6 a^(-2) b c^(-3) / (4 a b^(-3) c^2)) ÷ (5 a^(-3) b^2 c^(-1) / (3 a b^(-2) c^3)) and express the final answer using positive/negative exponents as needed.

Accepted Answer

Introduction / Context: This is a standard indices exercise involving negative exponents and division of monomials. The safest route is to simplify each fraction separately using index rules, then divide them. Carefully add or subtract exponents when multiplying or dividing like bases, and reduce numerical coefficients. Given Data / Assumptions: First factor: (6 a^(-2) b c^(-3)) / (4 a b^(-3) c^2). Second factor: (5 a^(-3) b^2 c^(-1)) / (3 a b^(-2) c^3). All variables represent nonzero real numbers (so divisions are valid). Concept / Approach: Use three rules repeatedly: a^m / a^n = a^(m−n); (ab)^m = a^m b^m; and handle coefficients separately. Simplify each fraction to a single monomial, then compute the quotient of those monomials by subtracting exponents and dividing coefficients. Step-by-Step Solution: First fraction: coefficients 6/4 = 3/2.a-exponent: (−2) − (1) = −3 ⇒ a^(−3).b-exponent: 1 − (−3) = 4 ⇒ b^4.c-exponent: (−3) − 2 = −5 ⇒ c^(−5).So F1 = (3/2) a^(−3) b^4 c^(−5).Second fraction: coefficients 5/3.a-exponent: (−3) − (1) = −4 ⇒ a^(−4).b-exponent: 2 − (−2) = 4 ⇒ b^4.c-exponent: (−1) − 3 = −4 ⇒ c^(−4).So F2 = (5/3) a^(−4) b^4 c^(−4).Quotient: F1 ÷ F2 = (3/2)/(5/3) * a^(−3 − (−4)) * b^(4−4) * c^(−5 − (−4)).Compute: (3/2)*(3/5) = 9/10; a-exponent = 1; b-exponent = 0; c-exponent = −1.Final: (9/10) a c^(−1) = (9/10) · a / c. Verification / Alternative check: Convert to positive-exponent form: 9a/(10c). Any equivalent representation matches the simplified result. Why Other Options Are Wrong: Options with c, c^2, or c^(−3) come from exponent slips; a^(−1) would require subtracting in reverse. Common Pitfalls: Mixing the direction of subtraction for exponents during division and forgetting to invert the second coefficient when dividing fractions. Final Answer: 9/10 · a c^-1

Question 4

Use exponent rules to condense powers of 3: Evaluate 27^3 × 3^4 ÷ 3^10 and choose the correct value.

Accepted Answer

Introduction / Context: This question tests consolidation of exponents using the same base. The key is to rewrite 27 as a power of 3 and then apply the rules for multiplying and dividing powers with the same base: add exponents when multiplying and subtract when dividing. Given Data / Assumptions: Expression: 27^3 × 3^4 ÷ 3^10. Recall: 27 = 3^3. Exponent rules: a^m * a^n = a^(m+n), and a^m / a^n = a^(m−n). Concept / Approach: Express all terms as powers of 3. Then use exponent laws to combine them into a single power of 3. Finally, evaluate that power or identify the corresponding integer. Step-by-Step Solution: Rewrite 27^3 as (3^3)^3 = 3^(9).Multiply: 3^9 × 3^4 = 3^(9+4) = 3^13.Divide by 3^10: 3^13 ÷ 3^10 = 3^(13−10) = 3^3.Evaluate: 3^3 = 27. Verification / Alternative check: You can also pair 3^4 ÷ 3^10 = 3^(−6) and then multiply 27^3 * 3^(−6) = 3^9 * 3^(−6) = 3^3 = 27. Why Other Options Are Wrong: 9 and 81 are 3^2 and 3^4 respectively—off by exponent arithmetic; 1/27 corresponds to 3^(−3), which would arise from reversing the subtraction; 3 would require exponent 1. Common Pitfalls: Forgetting to convert 27 into 3^3, or subtracting exponents in the wrong order when dividing. Final Answer: 27

Question 5

Combine exponents with a common base: If 10^(2/5) × 10^(8/5) = 10^n, determine the value of n.

Accepted Answer

Introduction / Context: This quick test checks your fluency with adding exponents when multiplying like bases. The base 10 is common to both terms, so the product is another power of 10 with an exponent equal to the sum of the original exponents. Given Data / Assumptions: Bases are the same: 10. Exponents: 2/5 and 8/5. Property: a^m * a^n = a^(m + n) for any real exponents m and n. Concept / Approach: Add the fractional exponents: 2/5 + 8/5 = 10/5. Then simplify the fraction to get an integer exponent. The result is a simple power of 10 that can be named or computed numerically if needed. Step-by-Step Solution: n = (2/5) + (8/5) = 10/5 = 2.Therefore 10^(2/5) × 10^(8/5) = 10^2 = 100. Verification / Alternative check: Evaluate numerically: 10^(0.4) × 10^(1.6) = 10^(2.0) by exponent addition, confirming n = 2. Why Other Options Are Wrong: 3, 4, 6, 10 reflect mis-additions or multiplications of exponents instead of addition. Common Pitfalls: Multiplying exponents (incorrect) instead of adding them when bases are equal. Final Answer: 2

Question 6

Apply the laws of indices to simplify variables and coefficients: Compute 12 x^2 y^4 z^3 ÷ (2 x y^2 × 3 y z^2) and express the answer in simplest monomial form.

Accepted Answer

Introduction / Context: This is a basic practice question on simplifying products and quotients of monomials. The idea is to reduce the numerical coefficient and then subtract exponents of like bases appearing in numerator and denominator. Given Data / Assumptions: Numerator: 12 x^2 y^4 z^3. Denominator: (2 x y^2) × (3 y z^2) = 6 x y^3 z^2. All variables x, y, z are nonzero real numbers. Concept / Approach: Divide coefficients: 12 ÷ 6 = 2. For variables, subtract exponents according to a^m / a^n = a^(m−n). Do this separately for x, y, and z. The result should be a single monomial with positive exponents. Step-by-Step Solution: Coefficient: 12 / 6 = 2.x: x^2 / x^1 = x^(2−1) = x.y: y^4 / y^3 = y^(4−3) = y.z: z^3 / z^2 = z^(3−2) = z.Final simplified form: 2 x y z. Verification / Alternative check: Substitute simple nonzero values (e.g., x = y = z = 1) to see that both sides equal 2, supporting the algebra. Why Other Options Are Wrong: xyz and 1/(xyz) ignore the coefficient change and/or exponent subtraction; “2 xyz” is equivalent in spacing to 2xyz (correct), but other distractors like x^2 y z reflect incorrect exponent arithmetic. Common Pitfalls: Forgetting to combine the two factors in the denominator first and mishandling exponent subtraction order when dividing. Final Answer: 2xyz

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