Question 1

If 3^(2x) × 3^(x−1) = 81, find x.

Accepted Answer

Step 1: Use index law a^m × a^n = a^(m+n): 3^(2x + x − 1) = 81. Step 2: Simplify: 3^(3x − 1) = 81. Step 3: Express 81 as power of 3: 81 = 3⁴. Step 4: So 3^(3x − 1) = 3⁴, which means 3x − 1 = 4. Step 5: Solve: 3x = 5, so x = 5/3.

Question 2

Simplify: (∛64)² ÷ (√16)³ × ∜81

Accepted Answer

Step 1: Evaluate ∛64 = 4, so (∛64)² = 4² = 16. Step 2: Evaluate √16 = 4, so (√16)³ = 4³ = 64. Step 3: Evaluate ∜81 = 3 (since 3⁴ = 81). Step 4: Calculate 16 ÷ 64 × 3 = (16/64) × 3 = (1/4) × 3 = 3/4 = 0.75. Therefore, the answer is 3/4 or 0.75.

Question 3

Rationalize the denominator: 5/√5

Accepted Answer

Step 1: Multiply numerator and denominator by √5: (5 × √5)/(√5 × √5). Step 2: Simplify: (5√5)/5. Step 3: Cancel: √5.

Question 4

Simplify: (2^5 × 2^3) ÷ 2^6

Accepted Answer

Step 1: Using the law of indices, 2^5 × 2^3 = 2^(5+3) = 2^8. Step 2: 2^8 ÷ 2^6 = 2^(8-6) = 2^2 = 4.

Question 5

If 2^(3x) = 512, find the value of 2^(x+1).

Accepted Answer

Step 1: Express 512 as a power of 2: 512 = 2⁹. Step 2: So 2^(3x) = 2⁹, which means 3x = 9. Step 3: Solve for x: x = 3. Step 4: Calculate 2^(x+1) = 2^(3+1) = 2⁴ = 16.

Question 6

Simplify: √(144) + ∛(125)

Accepted Answer

Step 1: Find √144 = 12 (since 12² = 144). Step 2: Find ∛125 = 5 (since 5³ = 125). Step 3: Add the results: 12 + 5 = 17. Therefore, the answer is 17.

Question 7

If 2^x = 32, find the value of x.

Accepted Answer

Step 1: Express 32 as a power of 2: 32 = 2^5. Step 2: Since 2^x = 2^5, we have x = 5.

Question 8

Simplify: √(50) + √(8) - √(18)

Accepted Answer

Step 1: √50 = √(25 × 2) = 5√2. Step 2: √8 = √(4 × 2) = 2√2. Step 3: √18 = √(9 × 2) = 3√2. Step 4: 5√2 + 2√2 - 3√2 = (5 + 2 - 3)√2 = 4√2.

Question 9

Evaluate: √(144) + √(64) - √(36)

Accepted Answer

Step 1: √144 = 12. Step 2: √64 = 8. Step 3: √36 = 6. Step 4: 12 + 8 - 6 = 14.

Question 10

Simplify: (3^4)^2 ÷ 3^5

Accepted Answer

Step 1: Using the power rule, (3^4)^2 = 3^(4×2) = 3^8. Step 2: 3^8 ÷ 3^5 = 3^(8-5) = 3^3 = 27.

Question 11

If 3^x = 27 and 2^y = 32, find x + y.

Accepted Answer

Step 1: Solve 3^x = 27. Since 27 = 3³, we have x = 3. Step 2: Solve 2^y = 32. Since 32 = 2⁵, we have y = 5. Step 3: Add: x + y = 3 + 5 = 8.

Question 12

Rationalize: 1/(∛2) and express in the form ∛a/b, where a and b are integers.

Accepted Answer

Step 1: We have 1/(∛2) = 1/2^(1/3). Step 2: Multiply numerator and denominator by 2^(2/3) to make the denominator a perfect cube: [1 × 2^(2/3)] / [2^(1/3) × 2^(2/3)] = 2^(2/3) / 2 = ∛4 / 2.

Question 13

Simplify: (16^(1/4) × 8^(1/3)) / 4^(1/2)

Accepted Answer

Step 1: Express all bases as powers of 2: 16 = 2⁴, 8 = 2³, 4 = 2². Step 2: Rewrite: (2⁴)^(1/4) × (2³)^(1/3) / (2²)^(1/2) = 2¹ × 2¹ / 2¹. Step 3: Simplify: 2 × 2 / 2 = 2.

Question 14

Simplify: (81)^(−3/4) × (27)^(2/3) ÷ (9)^(−1/2)

Accepted Answer

Step 1: Express all bases as powers of 3: 81 = 3⁴, 27 = 3³, 9 = 3². Step 2: Rewrite: (3⁴)^(−3/4) × (3³)^(2/3) ÷ (3²)^(−1/2) = 3^(−3) × 3² ÷ 3^(−1). Step 3: Simplify exponents: 3^(−3+2−(−1)) = 3^(−3+2+1) = 3⁰ = 1. Therefore, answer is 1.

Question 15

If 2^(3x) = 512, find the value of 2^(x−1).

Accepted Answer

Step 1: Express 512 as a power of 2: 512 = 2⁹. Step 2: So 2^(3x) = 2⁹, which means 3x = 9. Step 3: Solve for x: x = 3. Step 4: Calculate 2^(x−1) = 2^(3−1) = 2² = 4.

Question 16

If √(a) + √(b) = 7 and √(a) - √(b) = 1, find the value of ab.

Accepted Answer

Step 1: Let √(a) = p and √(b) = q. Then p + q = 7 and p - q = 1. Step 2: Add the two equations: (p+q) + (p-q) = 7 + 1, so 2p = 8, thus p = 4. Step 3: Subtract the second from the first: (p+q) - (p-q) = 7 - 1, so 2q = 6, thus q = 3. Step 4: Therefore √(a) = 4 and √(b) = 3, so a = 16 and b = 9. Step 5: ab = 16 × 9 = 144.

Question 17

Simplify: (∛(64) × ⁴√(81)) / (√(16) × ∛(27)). Express your answer as a single surd or integer.

Accepted Answer

Step 1: Convert each term to exponential form. ∛(64) = 64^(1/3) = (4^3)^(1/3) = 4. ⁴√(81) = 81^(1/4) = (3^4)^(1/4) = 3. √(16) = 16^(1/2) = 4. ∛(27) = 27^(1/3) = (3^3)^(1/3) = 3. Step 2: Substitute into the expression: (4 × 3) / (4 × 3) = 12/12 = 1.

Question 18

If 2^(x+1) + 2^(x+2) + 2^(x+3) = 224, find the value of 2^x.

Accepted Answer

Step 1: Factor out 2^x from the left side. 2^(x+1) + 2^(x+2) + 2^(x+3) = 2^x × 2^1 + 2^x × 2^2 + 2^x × 2^3 = 2^x(2 + 4 + 8) = 2^x × 14. Step 2: Set equal to 224: 2^x × 14 = 224. Step 3: Solve for 2^x: 2^x = 224/14 = 16. Step 4: Verify: 2^x = 16 means x = 4. Check: 2^5 + 2^6 + 2^7 = 32 + 64 + 128 = 224. ✓

Question 19

If 3^(2x-1) × 9^(x+1) = 27^(2x), find the value of x.

Accepted Answer

Step 1: Express all terms with the same base (base 3). 9 = 3^2, so 9^(x+1) = (3^2)^(x+1) = 3^(2(x+1)) = 3^(2x+2). 27 = 3^3, so 27^(2x) = (3^3)^(2x) = 3^(6x). Step 2: Rewrite the equation: 3^(2x-1) × 3^(2x+2) = 3^(6x). Step 3: Apply product rule on left side: 3^((2x-1)+(2x+2)) = 3^(6x). Step 4: Simplify exponent on left: 3^(4x+1) = 3^(6x). Step 5: Since bases are equal, exponents must be equal: 4x + 1 = 6x. Step 6: Solve for x: 1 = 6x - 4x = 2x, so x = 1/2.

Question 20

Simplify: (√8 + √18 - √32) / √2

Accepted Answer

Step 1: Simplify each surd separately. √8 = √(4×2) = 2√2. √18 = √(9×2) = 3√2. √32 = √(16×2) = 4√2. Step 2: Substitute into the expression: (2√2 + 3√2 - 4√2) / √2. Step 3: Combine like terms in numerator: (5√2 - 4√2) / √2 = √2 / √2. Step 4: Simplify: √2 / √2 = 1.

Question 21

If (∛5)^(2x) × (∛5)^(x+3) = 5^2, find x.

Accepted Answer

Step 1: Express ∛5 in exponential form: ∛5 = 5^(1/3). Step 2: Rewrite the equation: (5^(1/3))^(2x) × (5^(1/3))^(x+3) = 5^2. Step 3: Apply power rule: 5^(2x/3) × 5^((x+3)/3) = 5^2. Step 4: Apply product rule: 5^((2x/3) + ((x+3)/3)) = 5^2. Step 5: Simplify exponent on left: 5^((2x + x + 3)/3) = 5^2, which gives 5^((3x+3)/3) = 5^2. Step 6: Simplify further: 5^(x+1) = 5^2. Step 7: Equate exponents: x + 1 = 2, so x = 1.

Question 22

If √(x + √(x + √(x + ...))) = 3, find the value of x.

Accepted Answer

Step 1: Let y = √(x + √(x + √(x + ...))). This is an infinite nested radical. Step 2: Since the pattern repeats infinitely, the expression inside the first square root is also equal to y. Therefore: y = √(x + y). Step 3: We're given that y = 3. Substitute: 3 = √(x + 3). Step 4: Square both sides: 9 = x + 3. Step 5: Solve for x: x = 9 - 3 = 6.

Question 23

If 2^(3x) = 64, find the value of 2^(x+1).

Accepted Answer

Step 1: Rewrite 64 as a power of 2: 64 = 2⁶. Step 2: So 2^(3x) = 2⁶, which means 3x = 6. Step 3: Solve for x: x = 2. Step 4: Calculate 2^(x+1) = 2^(2+1) = 2³ = 8.

Question 24

What is the value of (√9 × √16) / √4?

Accepted Answer

Step 1: √9 = 3. Step 2: √16 = 4. Step 3: √4 = 2. Step 4: (3 × 4) / 2 = 12 / 2 = 6. Therefore, the answer is 6.

Question 25

Simplify: (3² × 3⁴) / 3³

Accepted Answer

Step 1: Using the law of indices, 3² × 3⁴ = 3^(2+4) = 3⁶. Step 2: 3⁶ / 3³ = 3^(6-3) = 3³. Step 3: 3³ = 27. Therefore, the answer is 27.

Question 26

If 5^x = 125, what is the value of x?

Accepted Answer

Step 1: Express 125 as a power of 5: 125 = 5³. Step 2: So 5^x = 5³. Step 3: Comparing both sides, x = 3. Therefore, the answer is 3.

Question 27

Simplify: ∛27 + ∛8 - ∛1

Accepted Answer

Step 1: ∛27 = 3 (since 3³ = 27). Step 2: ∛8 = 2 (since 2³ = 8). Step 3: ∛1 = 1 (since 1³ = 1). Step 4: 3 + 2 - 1 = 4. Therefore, the answer is 4.

Question 28

Simplify: (2³ × 2⁵) ÷ 2⁶

Accepted Answer

Step 1: Using the law of indices, 2³ × 2⁵ = 2^(3+5) = 2⁸. Step 2: 2⁸ ÷ 2⁶ = 2^(8-6) = 2². Step 3: 2² = 4. Therefore, the answer is 4.

Question 29

What is the value of (√5 + √3)²?

Accepted Answer

Step 1: Use the identity (a + b)² = a² + 2ab + b². Here a = √5 and b = √3. Step 2: a² = 5, b² = 3, and 2ab = 2 × √5 × √3 = 2√15. Step 3: Add: 5 + 3 + 2√15 = 8 + 2√15. Therefore, the answer is 8 + 2√15.

Question 30

What is the value of (√16 + √25) × √4?

Accepted Answer

Step 1: √16 = 4. Step 2: √25 = 5. Step 3: √4 = 2. Step 4: (4 + 5) × 2 = 9 × 2 = 18. Therefore, the answer is 18.

Question 31

If 5^(2x-1) = 125, find x.

Accepted Answer

Step 1: Rewrite 125 as a power of 5: 125 = 5³. Step 2: So 5^(2x-1) = 5³, which means 2x - 1 = 3. Step 3: Solve for x: 2x = 4, therefore x = 2.

Question 32

Simplify: (∛8 × ∜16) / ∛27

Accepted Answer

Step 1: ∛8 = ∛(2³) = 2. Step 2: ∜16 = ∜(2⁴) = 2. Step 3: ∛27 = ∛(3³) = 3. Step 4: Substitute: (2 × 2) / 3 = 4/3.

Question 33

If 2^(3x) = 512, find the value of 2^(x−1).

Accepted Answer

Step 1: Express 512 as a power of 2: 512 = 2⁹. Step 2: So 2^(3x) = 2⁹, which gives 3x = 9, thus x = 3. Step 3: Calculate 2^(x−1) = 2^(3−1) = 2² = 4.

Question 34

Simplify: √(√256) + ∛(∛512) − ⁴√(⁴√4096)

Accepted Answer

Step 1: √(√256) = (256)^(1/4) = (2⁸)^(1/4) = 2² = 4. Step 2: ∛(∛512) = (512)^(1/9) = (2⁹)^(1/9) = 2¹ = 2. Step 3: ⁴√(⁴√4096) = (4096)^(1/16) = (2¹⁶)^(1/16) = 2¹ = 2. Step 4: 4 + 2 − 2 = 4.

Question 35

Rationalize: 1/(2 + √3) and express in the form (a − b√3) where a and b are integers.

Accepted Answer

Step 1: Multiply numerator and denominator by (2 − √3): [1 × (2 − √3)] / [(2 + √3)(2 − √3)]. Step 2: Denominator = 4 − 3 = 1. Step 3: Numerator = 2 − √3. Step 4: Result = 2 − √3, so a = 2, b = 1.

Question 36

If x = 2^(1/3) + 2^(-1/3), then find the value of x³ - 3x.

Accepted Answer

Step 1: Let a = 2^(1/3) and b = 2^(-1/3). Note that ab = 2^(1/3) × 2^(-1/3) = 2^0 = 1. Step 2: We have x = a + b. Step 3: Cube both sides: x³ = (a + b)³ = a³ + 3a²b + 3ab² + b³. Step 4: Simplify: x³ = a³ + b³ + 3ab(a + b) = 2 + 1/2 + 3(1)(x) = 2.5 + 3x. Step 5: Therefore x³ - 3x = 2.5 = 5/2.

Question 37

If √(2 + √3) = (√a + √b) / √2, where a and b are positive integers, find a + b.

Accepted Answer

Step 1: Square both sides: 2 + √3 = (a + b + 2√(ab)) / 2. Step 2: Multiply both sides by 2: 4 + 2√3 = a + b + 2√(ab). Step 3: Compare rational and irrational parts: a + b = 4 and 2√(ab) = 2√3. Step 4: From the second equation: √(ab) = √3, so ab = 3. Step 5: Solve the system: a + b = 4 and ab = 3. These are roots of t² - 4t + 3 = 0, which factors as (t - 1)(t - 3) = 0. Step 6: So a = 3 and b = 1 (or vice versa). Step 7: Therefore a + b = 4.

Question 38

If √(2 + √3) = √a + √b, where a and b are rational numbers, then find a + b.

Accepted Answer

Step 1: Square both sides: 2 + √3 = (√a + √b)² = a + b + 2√(ab). Step 2: Comparing rational and irrational parts: a + b = 2 and 2√(ab) = √3. Step 3: From 2√(ab) = √3, square both sides: 4ab = 3, so ab = 3/4. Step 4: Now a and b are roots of the quadratic t² − 2t + 3/4 = 0. Step 5: Using the quadratic formula: t = [2 ± √(4 − 3)]/2 = [2 ± 1]/2. So t = 3/2 or t = 1/2. Step 6: Therefore, a = 3/2 and b = 1/2 (or vice versa). Step 7: a + b = 3/2 + 1/2 = 2.

Question 39

If (2^a × 3^b × 5^c)^(1/6) = 2^(1/3) × 3^(1/2) × 5^(1/6), then find the value of a + b + c.

Accepted Answer

Step 1: Expand the left side using exponent rules: 2^(a/6) × 3^(b/6) × 5^(c/6) = 2^(1/3) × 3^(1/2) × 5^(1/6). Step 2: Compare exponents of like bases. For base 2: a/6 = 1/3, so a = 2. For base 3: b/6 = 1/2, so b = 3. For base 5: c/6 = 1/6, so c = 1. Step 3: a + b + c = 2 + 3 + 1 = 6.

Question 40

If (2^a × 3^b × 5^c)^(1/6) = 2^(1/3) × 3^(1/2) × 5^(1/6), then find the value of (a + b + c).

Accepted Answer

Step 1: Expand the left side: 2^(a/6) × 3^(b/6) × 5^(c/6) = 2^(1/3) × 3^(1/2) × 5^(1/6). Step 2: Compare exponents of like bases. For base 2: a/6 = 1/3, so a = 2. For base 3: b/6 = 1/2, so b = 3. For base 5: c/6 = 1/6, so c = 1. Step 3: a + b + c = 2 + 3 + 1 = 6.

Question 41

If 3^x = 5^y = 15^z, then which of the following is true?

Accepted Answer

Step 1: Let 3^x = 5^y = 15^z = k (some constant). Step 2: Then x = log_3(k), y = log_5(k), z = log_15(k). Step 3: Using change of base: x = ln(k)/ln(3), y = ln(k)/ln(5), z = ln(k)/ln(15). Step 4: Note that 15 = 3 × 5, so ln(15) = ln(3) + ln(5). Step 5: Therefore 1/z = [ln(3) + ln(5)]/ln(k) = ln(3)/ln(k) + ln(5)/ln(k) = 1/x + 1/y. Step 6: Thus 1/z = 1/x + 1/y.

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