Powers Exponents Surds Competitive Exams
50 questions · Mathematics
-
Question 1 of 50
Simplify: 1/(√2 + 1) + 1/(√3 + √2) + 1/(2 + √3).
ExplanationRationalising gives (√2 − 1) + (√3 − √2) + (2 − √3). The terms cancel, leaving 1.
-
Question 2 of 50
Simplify √72.
Explanation√72 = √(36 × 2) = 6√2.
-
Question 3 of 50
Simplify (√5 + 2)(√5 − 2).
Explanation(a + b)(a − b) = a² − b² = 5 − 4 = 1.
-
Question 4 of 50
Find the value of (2⁻¹ + 3⁻¹)⁻¹.
Explanation2⁻¹ + 3⁻¹ = 1/2 + 1/3 = 5/6. Its inverse is 6/5.
-
Question 5 of 50
Find the value of √(7 + 4√3).
Explanation(2 + √3)² = 4 + 3 + 4√3 = 7 + 4√3, so the square root is 2 + √3.
-
Question 6 of 50
The last two digits of 5¹⁰ are:
ExplanationAny power of 5 with exponent 2 or more ends in 25 (5² = 25, 5³ = 125, …).
-
Question 7 of 50
Solve for x: √(x + 7) = 5.
ExplanationSquare both sides: x + 7 = 25, so x = 18.
-
Question 8 of 50
Find the unit digit of 13⁵⁷.
ExplanationOnly the unit digit 3 matters. 57 ÷ 4 leaves remainder 1, so the digit is 3¹ = 3.
-
Question 9 of 50
Simplify (√3 + √2)².
Explanation(a + b)² = a² + 2ab + b² = 3 + 2√6 + 2 = 5 + 2√6.
-
Question 10 of 50
Rationalise 1/(√3 − √2).
ExplanationMultiply by (√3 + √2)/(√3 + √2): the denominator becomes 3 − 2 = 1, so the answer is √3 + √2.
-
Question 11 of 50
Find the unit digit of 9⁹⁹.
ExplanationPowers of 9 end in 9 for odd exponents and 1 for even exponents. 99 is odd, so the digit is 9.
-
Question 12 of 50
Simplify 2/(√3 + 1).
ExplanationMultiply by (√3 − 1)/(√3 − 1): the denominator is 3 − 1 = 2, so 2(√3 − 1)/2 = √3 − 1.
-
Question 13 of 50
Find the value of (0.25)^(−1/2).
Explanation0.25 = 1/4. (1/4)^(−1/2) = 4^(1/2) = 2.
-
Question 14 of 50
Find the value of √48 ÷ √3.
Explanation√48 ÷ √3 = √(48/3) = √16 = 4.
-
Question 15 of 50
Simplify: 9³ ÷ 3⁴.
Explanation9³ = 3⁶, so 3⁶ ÷ 3⁴ = 3² = 9.
-
Question 16 of 50
Find the value of (−2)⁻³.
Explanation(−2)⁻³ = 1/(−2)³ = 1/(−8) = −1/8.
-
Question 17 of 50
Find the value of √(12 + √(12 + √(12 + …))).
ExplanationLet the value be x. Then x = √(12 + x), so x² − x − 12 = 0, which gives x = 4 (the negative root is rejected).
-
Question 18 of 50
The surd ∛7 is a surd of order:
ExplanationThe order of a surd is the index of the root. In ∛7 the index is 3.
-
Question 19 of 50
Find the unit digit of 7¹²³.
ExplanationUnit digits of powers of 7 repeat in a cycle of 4 (7, 9, 3, 1). 123 ÷ 4 leaves remainder 3, so the digit is that of 7³ = 343, i.e. 3.
-
Question 20 of 50
Find the value of (2³)².
Explanation(aᵐ)ⁿ = aᵐⁿ, so (2³)² = 2⁶ = 64.
-
Question 21 of 50
Rationalise 1/(√5 + 2).
ExplanationMultiply by (√5 − 2)/(√5 − 2): the denominator becomes 5 − 4 = 1, so the answer is √5 − 2.
-
Question 22 of 50
Find the value of 2⁻³.
Explanationa⁻ⁿ = 1/aⁿ, so 2⁻³ = 1/2³ = 1/8.
-
Question 23 of 50
Find the unit digit of 2¹⁰⁰.
ExplanationPowers of 2 have unit digits 2, 4, 8, 6 in a cycle of 4. 100 is a multiple of 4, so the digit is 6.
-
Question 24 of 50
Simplify: 5⁷ ÷ 5⁴.
ExplanationSame base, subtract powers: 5⁷⁻⁴ = 5³ = 125.
-
Question 25 of 50
If 2ˣ = 1/16, then x equals:
Explanation1/16 = 2⁻⁴, so x = −4.
-
Question 26 of 50
Find the value of 16^(3/4).
Explanation16^(1/4) = 2, so 16^(3/4) = 2³ = 8.
-
Question 27 of 50
If 5²ˣ⁻¹ = 125, then x equals:
Explanation125 = 5³, so 2x − 1 = 3, which gives x = 2.
-
Question 28 of 50
Find the unit digit of the product 153 × 567 × 89.
ExplanationMultiply the unit digits: 3 × 7 = 21 → 1; 1 × 9 = 9. So the unit digit is 9.
-
Question 29 of 50
Simplify √12 + √27.
Explanation√12 = 2√3 and √27 = 3√3, so the sum is 5√3.
-
Question 30 of 50
Find the unit digit of 4⁷⁷.
ExplanationPowers of 4 end in 4 for odd exponents and 6 for even exponents. 77 is odd, so the digit is 4.
-
Question 31 of 50
Which is the greatest among √2, ∛3, ⁴√5 and ⁶√10?
ExplanationRaise each to the power 12 (the LCM of the indices): 2⁶ = 64, 3⁴ = 81, 5³ = 125 and 10² = 100. The largest is 125, so ⁴√5 is greatest.
-
Question 32 of 50
Find the value of 27^(−2/3).
Explanation27^(1/3) = 3, so 27^(2/3) = 9 and 27^(−2/3) = 1/9.
-
Question 33 of 50
Find the value of 5⁰ + 7⁰ + 9⁰.
ExplanationAny non-zero number raised to the power 0 equals 1, so the sum is 1 + 1 + 1 = 3.
-
Question 34 of 50
Which of the following laws of exponents is correct?
ExplanationThe power of a power multiplies the exponents. Products add exponents and quotients subtract them.
-
Question 35 of 50
Simplify: (4³ × 2⁴) ÷ 8².
Explanation4³ = 2⁶, 8² = 2⁶, so the expression = (2⁶ × 2⁴) ÷ 2⁶ = 2⁴ = 16.
-
Question 36 of 50
Find the value of (1/3)⁻².
Explanation(1/a)⁻ⁿ = aⁿ, so (1/3)⁻² = 3² = 9.
-
Question 37 of 50
Find the value of 125^(1/3) × 4^(1/2).
Explanation125^(1/3) = 5 and 4^(1/2) = 2, so the product is 10.
-
Question 38 of 50
Find the value of 3√5 × 2√5.
ExplanationMultiply coefficients and radicands: 3 × 2 × (√5 × √5) = 6 × 5 = 30.
-
Question 39 of 50
If 2ˣ = 64, then x equals:
Explanation64 = 2⁶, so x = 6.
-
Question 40 of 50
Find the unit digit of 2²⁰²³.
ExplanationThe cycle is 2, 4, 8, 6. 2023 ÷ 4 leaves remainder 3, so the digit is 8.
-
Question 41 of 50
If 3ˣ⁺¹ = 81, then x equals:
Explanation81 = 3⁴, so x + 1 = 4 and x = 3.
-
Question 42 of 50
Find the value of 99².
Explanation99² = (100 − 1)² = 10000 − 200 + 1 = 9801.
-
Question 43 of 50
Rationalise the denominator of 1/√2.
ExplanationMultiply top and bottom by √2: (1 × √2)/(√2 × √2) = √2/2.
-
Question 44 of 50
Find the value of 8^(2/3).
Explanation8^(1/3) = 2, so 8^(2/3) = 2² = 4.
-
Question 45 of 50
Simplify: 3⁴ × 3².
ExplanationSame base, add powers: 3⁴ × 3² = 3⁶ = 729.
-
Question 46 of 50
Find the unit digit of 3⁴⁷.
ExplanationUnit digits of powers of 3 cycle as 3, 9, 7, 1. 47 ÷ 4 leaves remainder 3, so the digit is 7.
-
Question 47 of 50
Which of the following is a surd?
Explanation√16 = 4, √25 = 5 and √49 = 7 are rational. √7 is irrational, so it is a surd.
-
Question 48 of 50
If x = √2 + 1, then the value of x + 1/x is:
Explanation1/x = 1/(√2 + 1) = √2 − 1. So x + 1/x = (√2 + 1) + (√2 − 1) = 2√2.
-
Question 49 of 50
Simplify √50.
Explanation√50 = √(25 × 2) = 5√2.
-
Question 50 of 50
Find the value of √18 × √2.
Explanation√18 × √2 = √36 = 6.
0 / 50 answered · 0 skipped
You can practise as a guest, but sign in to save your score and track your progress.