Laws of Exponents for Real NumbersPractice Questions & MCQs

Q1. Solve 4^x - 6·2^x + 8 = 0 and find the sum of all real values of x.

• A.3
• B.2
• C.6
• D.4

Answer: A. 3

Diagnose the question type - a typical math numerical. The data on the table: 4, 6, 183, 2, 8. We are after find the sum of all real values of x. Tool of choice - let t=2^x. Rearrange it for the unknown before substituting. Numbers in: let t=2^x → t^2-6t+8=0 factors to t=2,4 → x=1,2 → sum = 3. Lock in option A) 3. Trap-watch: option B) '2' fails since There are two solutions; add them; option C) '6' is wrong because Put t=2^x: t^2-6t+8=0.

Q2. If 2^a = (4)^p = (8)^r and a = 24, find p + r.

• A.20
• B.12
• C.8
• D.21

Answer: A. 20

Q3. If 2^a = (4)^p = (8)^r and a = 18, find p + r.

• A.15
• B.9
• C.6
• D.16

Answer: A. 15

Reading off the stem: 2; 4; 8; 18. The unknown asked is: find p + r. Relation we use - 2^18 = 2^(2p). This is the equation that links the given quantities to the unknown (math, chapter 'Number Systems'). Substituting: 2^18 = 2^(2p) → p = 18/2 = 9 → = 2^(3r) → r = 18/3 = 6. Thus the answer is A) 15. The other choices: option B) '9' fails since (4)^p = 2^(2p); 2p = 18; also 3r=18; option C) '6' is incorrect: Add p and r; option D) '16' misses the point - p = 9, r = 6; p+r = 15.