Question 1

If 3(x − y) = 27 and 3(x + y) = 243, find x.

Accepted Answer

Given data 3(x − y) = 27 ⇒ x − y = 9. 3(x + y) = 243 ⇒ x + y = 81. Concept / Approach Solve the system by adding to eliminate y. Step-by-step calculation (x − y) + (x + y) = 9 + 81 ⇒ 2x = 90 ⇒ x = 45. Verification Then y = 81 − 45 = 36. Check: x − y = 9 ✔. Common pitfalls Dividing by 3 after already simplifying; double-counting factors. Final Answer 45.

Question 2

(25)^{7.5} × (5)^{2.5} ÷ (125)^{1.5} equals 5 raised to which power?

Accepted Answer

Given data & idea Rewrite all bases as powers of 5: 25 = 5^2, 125 = 5^3. Use (a^m)^n = a^{mn} and combine exponents. Step-by-step calculation (25)^{7.5} = (5^2)^{7.5} = 5^{15}(5)^{2.5} = 5^{2.5}(125)^{1.5} = (5^3)^{1.5} = 5^{4.5}Overall exponent = 15 + 2.5 − 4.5 = 13. Verification Result = 5^{13} ⇒ power is 13. Common pitfalls Mistakenly adding denominators instead of subtracting exponents for division. Final Answer Power = 13.

Question 3

Simplify (x^b)^{(b + c − a)} · (x^c)^{(c + a − b)} · (x^a)^{(a + b − c)} to a single power of x.

Accepted Answer

Given data Expression: (x^b)^{(b + c − a)} · (x^c)^{(c + a − b)} · (x^a)^{(a + b − c)} Concept / Approach Use (x^p)^q = x^{pq}. When multiplying same bases, add exponents. Step-by-step exponent aggregation Exponent of x = b(b + c − a) + c(c + a − b) + a(a + b − c)= b^2 + bc − ab + c^2 + ca − bc + a^2 + ab − acCross terms cancel: (bc − bc), (−ab + ab), (ca − ac) ⇒ remaining = a^2 + b^2 + c^2. Verification Pick a = 1, b = 2, c = 3: exponent sum = 1 + 4 + 9 = 14; numeric expansion confirms. Common pitfalls Forgetting to distribute negatives in (b + c − a), etc. Final Answer Simplified form: x^{a^2 + b^2 + c^2}.

Question 4

If m and n are whole numbers with m^n = 121, evaluate (m − 1)^n + 1.

Accepted Answer

Given data m^n = 121. Concept / Approach Prime factorize 121 to find integer m and n. Step-by-step calculation 121 = 11^2 ⇒ m = 11, n = 2 (whole numbers).(m − 1)^n + 1 = (11 − 1)^2 + 1 = 10^2 + 1 = 101. Verification No other whole-number power representations exist for 121. Common pitfalls Interpreting mn as m×n rather than m^n; that would not give a unique value. Final Answer 101.

Question 5

Given 10^{0.48} = x, 10^{0.70} = y, and x^z = y^2, find z (approximate).

Accepted Answer

Given data x = 10^{0.48}, y = 10^{0.70}, and x^z = y^2. Concept / Approach Use logarithms/exponent rules: (10^{a})^{b} = 10^{ab}. Step-by-step calculation x^z = (10^{0.48})^{z} = 10^{0.48z}y^2 = (10^{0.70})^{2} = 10^{1.40}Equate exponents: 0.48z = 1.40 ⇒ z = 1.40 / 0.48 = 140/48 = 35/12 ≈ 2.92. Verification (10^{0.48})^{35/12} = 10^{(0.48×35/12)} = 10^{1.40} = (10^{0.70})^2. Common pitfalls Adding 0.48 and 0.70 or squaring incorrectly; remember to equate exponents. Final Answer 2.92 (approximately).

Question 6

If (a/b)^{x − 1} = (b/a)^{x − 3}, find x.

Accepted Answer

Given data (a/b)^{x − 1} = (b/a)^{x − 3}. Concept / Approach Use (b/a) = (a/b)^{−1} to rewrite both sides with the same base. Step-by-step simplification Right side: (b/a)^{x − 3} = ( (a/b)^{−1} )^{x − 3} = (a/b)^{−(x − 3)}.Equate exponents (same positive base a/b): x − 1 = −(x − 3).x − 1 = −x + 3 ⇒ 2x = 4 ⇒ x = 2. Verification Left: (a/b)^{1}; Right: (b/a)^{−1} = (a/b)^{1} ⇒ equal. Common pitfalls Treating (b/a) as independent base without converting; leads to messy logs. Final Answer 2.

Question 7

Evaluate (243)^{n/5} × 3^{2n + 1} ÷ (9^{n} × 3^{n − 1}).

Accepted Answer

Given data Expression: (243)^{n/5} × 3^{2n+1} / (9^{n} × 3^{n−1}). Concept / Approach Rewrite all as powers of 3 and then subtract exponents for division. Step-by-step calculation 243 = 3^5 ⇒ (243)^{n/5} = (3^5)^{n/5} = 3^{n}.Numerator exponent: n + (2n + 1) = 3n + 1 ⇒ 3^{3n + 1}.Denominator: 9^{n} × 3^{n − 1} = (3^2)^{n} × 3^{n − 1} = 3^{2n} × 3^{n − 1} = 3^{3n − 1}.Overall = 3^{(3n + 1) − (3n − 1)} = 3^{2} = 9. Verification Try n = 1: (243)^{0.2}×3^{3}/(9×3^{0}) = 3×27/9 = 9 ✔. Common pitfalls Forgetting to subtract exponents when dividing powers with the same base. Final Answer 9.

Question 8

The expression shown is incomplete: '1 + 1 = ? 1 + a(n − m) 1 + a(m − n)'. Based on the given text, the value cannot be determined.

Accepted Answer

Cannot be determined from the given information

Question 9

Evaluate: (10)^150 ÷ (10)^146

Accepted Answer

Concept / Approach Use the law of exponents: a^m ÷ a^n = a^(m − n) for a ≠ 0. Step-by-step calculation(10)^150 ÷ (10)^146 = 10^(150 − 146)= 10^4 = 10000 Quick checkSubtracting exponents is valid since the base (10) is the same. Final Answer10000

Question 10

If 5^a = 3125, find the value of 5^(a − 3).

Accepted Answer

Given data 5^a = 3125 Concept / Approach Express 3125 as a power of 5 and then reduce the exponent. Step-by-step calculation3125 = 5^5 ⇒ a = 55^(a − 3) = 5^(5 − 3) = 5^2 = 25 Common pitfalls Confusing 5a with 5^a. The problem states an exponential equation 5^a = 3125. Final Answer25

Question 11

The expression appears incomplete: '1 + 1 + 1 = ? 1 + x(b − a) + x(c − a) 1 + x(a − b) + x(c − b) 1 + x(b − c) + x(a − c)'. The value cannot be determined from the given text.

Accepted Answer

Cannot be determined from the given information

Question 12

Compute: (256)^0.16 × (256)^0.09

Accepted Answer

Concept / Approach For the same base, add exponents when multiplying: a^m × a^n = a^(m + n). Convert the result using prime factorization of the base. Step-by-step calculation(256)^0.16 × (256)^0.09 = 256^(0.16 + 0.09) = 256^0.25256 = 2^8 ⇒ 256^0.25 = (2^8)^(1/4) = 2^(8/4) = 2^2 = 4 Final Answer4

Question 13

Evaluate: (0.04)^(−1.5)

Accepted Answer

Concept / Approach a^(−k) = 1 / a^k, and fractional exponents correspond to roots. Rewrite 0.04 in fractional form. Step-by-step calculation0.04 = 4/100 = 1/25 = 25^(−1)(0.04)^(−1.5) = (25^(−1))^(−1.5) = 25^(1.5)25^(3/2) = (√25)^3 = 5^3 = 125 Final Answer125

Question 14

The expression '(17)^3.5 × (17)^? = 178' is inconsistent with integer-power arithmetic; the missing exponent cannot be resolved uniquely from the provided text.

Accepted Answer

Cannot be determined from the given information

Question 15

If x = 3 + 2√2, find √x.

Accepted Answer

Concept / Approach Try to express x as a perfect square of the form (√2 ± 1)^2. Step-by-step calculation(√2 + 1)^2 = (√2)^2 + 2·√2·1 + 1^2 = 2 + 2√2 + 1 = 3 + 2√2Therefore, x = (√2 + 1)^2 ⇒ √x = √2 + 1 Verification / Alternative(√2 − 1)^2 = 2 − 2√2 + 1 = 3 − 2√2 ≠ x, so the positive root is √2 + 1. Final Answer√2 + 1

Question 16

Evaluate the surd expression: Compute (√13)^4 (write your answer as an integer).

Accepted Answer

Introduction / Context:This is a straightforward surds-and-indices computation. Recognize how exponents distribute over radicals and simplify to an integer using exponent rules. No approximation is required. Given Data / Assumptions: Expression: (√13)^4. Use index and radical rules: √13 = 13^(1/2). Concept / Approach:Convert the radical to an exponent, then multiply exponents: (a^(m))^(n) = a^(m n). This yields a power of 13 that is easy to evaluate exactly. Step-by-Step Solution: (√13)^4 = (13^(1/2))^4 = 13^(1/2 * 4) = 13^2.13^2 = 169. Verification / Alternative check:Square first, then square again: (√13)^2 = 13; 13^2 = 169. Same result. Why Other Options Are Wrong: 520, 14280: Products unrelated to the correct exponent evaluation. 28561: That is 13^4; we have 13^2. 13: That equals (√13)^2, not to the 4th power. Common Pitfalls:Forgetting that exponents multiply when raising a power to a power, or misreading the expression as √(13^4). Final Answer:169

Question 17

Simplify the indices expression: Compute 16^(3/2) + 16^(−3/2) and express your answer as a simplified fraction.

Accepted Answer

Introduction / Context:Rational exponents allow us to rewrite powers in terms of roots. 16^(3/2) is the cube of the square root (or the square root of the cube) of 16. Similarly, negative exponents indicate reciprocals. Combine these properties to obtain an exact fractional sum. Given Data / Assumptions: 16^(3/2) and 16^(−3/2). 16 = 2^4, √16 = 4. Concept / Approach:Compute each term separately: 16^(3/2) = (16^(1/2))^3 = 4^3 = 64. For the negative exponent, 16^(−3/2) = 1 / 16^(3/2) = 1/64. Then add the two values and convert to a single fraction. Step-by-Step Solution: 16^(3/2) = 64.16^(−3/2) = 1/64.Sum = 64 + 1/64 = (64*64 + 1) / 64 = 4097/64. Verification / Alternative check:Compute numerically: 64 + 0.015625 = 64.015625 = 4097/64. Matches. Why Other Options Are Wrong: 0, 1, 16/9097: Incorrect evaluations of the exponents or inversion. 65/64: Misses the large 16^(3/2) term (64). Common Pitfalls:Confusing 16^(3/2) with (16^3)^(1/2) without carefully evaluating; both are 64, but ensure steps are consistent. Do not forget the reciprocal for the negative exponent. Final Answer:4097/64

Question 18

Evaluate a power-of-a-power ratio: Simplify the expression [(12)^(−2)]^2 ÷ [(12)^2]^(−2) and give the exact value.

Accepted Answer

Introduction / Context:This tests exponent laws, specifically (a^m)^n = a^(m n), and how negative exponents invert the base. Carefully apply the rules to both numerator and denominator before simplifying the ratio of equal powers. Given Data / Assumptions: Expression: [(12)^(−2)]^2 ÷ [(12)^2]^(−2). Use standard laws of indices. Concept / Approach:Compute the power of a power for both parts, reduce each to a single power of 12, then divide. Identical powers in numerator and denominator will cancel to 1. Step-by-Step Solution: Numerator: [(12)^(−2)]^2 = 12^(−4).Denominator: [(12)^2]^(−2) = 12^(2 * −2) = 12^(−4).Ratio = 12^(−4) / 12^(−4) = 12^(−4 − (−4)) = 12^0 = 1. Verification / Alternative check:Assign 12^(−4) = 1/12^4 for both; clearly their quotient is 1. Why Other Options Are Wrong: 12, 4.8, 12/144, 1/12: Result from arithmetic slips or misapplying negative exponents. Common Pitfalls:Forgetting that raising to a negative exponent inverts, and that dividing equal powers subtracts exponents. Final Answer:1

Question 19

Combine exponents of the same base: Compute the missing exponent k in (16)^9 ÷ (16)^4 × (16)^3 = (16)^k.

Accepted Answer

Introduction / Context:When multiplying or dividing powers with the same base, exponents add or subtract respectively. This is a direct application of the indices laws that allows consolidation to a single power of the base 16. Given Data / Assumptions: Expression: 16^9 ÷ 16^4 × 16^3 = 16^k. All operations are on the same base (16). Concept / Approach:Use two rules: a^m ÷ a^n = a^(m − n) and a^m × a^n = a^(m + n). Apply them in sequence to combine all powers into a single exponent k. Step-by-Step Solution: First division: 16^9 ÷ 16^4 = 16^(9 − 4) = 16^5.Then multiply by 16^3: 16^5 × 16^3 = 16^(5 + 3) = 16^8.So k = 8. Verification / Alternative check:Rearranging the order (multiplication before division) still yields 9 − 4 + 3 = 8, confirming consistency. Why Other Options Are Wrong: 10, 12: Result from adding all exponents or forgetting to subtract for division. 6.75, 7: Non-integer exponents not supported by the given integer-exponent operations. Common Pitfalls:Applying multiplication and division in the wrong way to the exponents. Keep track: divide ⇒ subtract; multiply ⇒ add. Final Answer:8

Question 20

Evaluate the expression using index laws and real-number arithmetic: [(11)^3 × (6)^2] ÷ (4)^3 = ?

Accepted Answer

Introduction / Context:This problem tests basic laws of indices and order of operations. We must carefully compute powers first, then multiply and divide. Converting to exact integers as long as possible helps prevent rounding errors, and only at the end should we convert to decimals if needed.Given Data / Assumptions: Compute (11)^3, (6)^2, and (4)^3. Multiply 11^3 and 6^2, then divide by 4^3. All quantities are real and positive. Concept / Approach:Use the exponent definitions a^m * a^n = a^(m+n) (when bases match) and evaluate each power separately when bases differ. Respect BODMAS/PEMDAS: powers → multiplication → division → addition/subtraction.Step-by-Step Solution: 11^3 = 11 * 11 * 11 = 1331.6^2 = 36.4^3 = 64.Multiply numerator: 1331 * 36 = 47916.Divide: 47916 ÷ 64 = 748.6875. Verification / Alternative check:Break 36 as 64 * (36/64) = 64 * 0.5625; then (1331 * 36)/64 = 1331 * 0.5625 = 748.6875, confirming the previous result without long division.Why Other Options Are Wrong: 2994.75 and 4492.125: These are results of multiplying correctly but forgetting to divide by 64 or dividing by a different number. 272.25: Too small; suggests dividing by an extra factor or miscomputing 11^3. Common Pitfalls:Typical mistakes include computing 11^3 as 121 or 133 instead of 1331, or dividing by 4 rather than 4^3. Always compute exponents before other operations. Final Answer: 748.6875

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