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 check Subtracting exponents is valid since the base (10) is the same. Final Answer 10000

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 calculation 3125 = 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 Answer 25

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 Answer 4

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 calculation 0.04 = 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 Answer 125

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

Solve the below equation and find the value of ?. √(13)4 =?

Accepted Answer

? = √(13)4 = √13 x 13 x 13 x 13 Apply the rule of square root. = 13 x 13 = 169

Question 17

163/2 + 16 -3/2 =?

Accepted Answer

163/2 + 16-3/2 Apply the law of Fractional Exponents and Laws of Exponents am x an = am+n a-m = 1/am ⇒163/2 + 1/163/2 ⇒(161/2)3 + 1/(161/2)3 ⇒(42 x 1/2)3 + 1/(42 x 1/2)3 ⇒43 + 1/43 ⇒ 64 + 1/64 ⇒ (64 x 64+ 1)/64 = (4096+1)/64 = 4097/64

Question 18

[(12)-2]2/[(12)2]-2 =?

Accepted Answer

? = [(12)-2]2/[(12)2]-2 Apply the Algebra Law, (am)n = amn = (12) - 4/(12) - 4 = 1

Question 19

(16)9 ÷ (16)4 x (16)3 = (16)?

Accepted Answer

Given that ,(16)9 ÷ (16)4 x (16) 3 = (16)? Apply the law of Fractional Exponents and Laws of Exponents (am) x (an) = am + n-------------------1 am ÷ an = am ? n----------------2 ⇒ (16)9 ÷ (16)4 x (16) 3 = (16)? ⇒ (16)? = (16)9 + 3 - 4 ⇒ ? = 12 - 4 = 8

Question 20

[ (11)3 x (6)2] ÷ (4)3 =?

Accepted Answer

? = [ (11)3 x (6)2 ] x 1/43 ? = [ 113 x 62 ] x 1/43 Solve the equation by algebra law. ?= [ 1331 x 36 ] x 1/64 =1331 x 36 / 64 ? = 748.6875

Surds and Indices Questions