Practice Questions
Concept Explanation
Indices (exponents) follow laws of powers. Surds are irrational roots, usually square roots, simplified using root laws.
Index Laws
- am × an = am+n
- am ÷ an = am−n
- (am)n = amn
- a0 = 1 (a ≠ 0); a−n = 1/an
Surd Laws
- √(ab) = √a × √b
- √(a/b) = √a / √b
- Rationalization: multiply numerator and denominator by the conjugate surd.
Example
1/(√5 − 2) = (√5 + 2) / [(√5)² − 2²] = √5 + 2.
Common Trap
√(a + b) ≠ √a + √b. Do not split sums under a root.
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Q1 Easy
What is the value of 2³ × 2&sup4; when written as a single power of 2?
Explanation23+4 = 2&sup7;. -
Q2 Easy
What is 5&sup6; ÷ 5² equal to when expressed as a single power of 5?
Explanation56−2 = 5&sup4;. -
Q3 Easy
What is the numerical value of 3&sup0; + 2¹?
Explanation1 + 2 = 3. -
Q4 Easy
Simplify: √50.
Explanation√50 = √(25 × 2) = 5√2. -
Q5 Medium
What is √12 + √27 equal to when both surds are fully simplified and combined?
Explanation2√3 + 3√3 = 5√3. -
Q6 Medium
What is the value of 2⊃−3;?
Explanation2⊃−3; = 1/2³ = 1/8. -
Q7 Easy
What is (√7)² equal to?
Explanation(√7)² = 7.
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Q8 Medium
When the denominator of 1/(√3) is rationalized by multiplying numerator and denominator by √3, what is the resulting expression?
ExplanationMultiply by √3/√3: √3/3. -
Q9 Medium
What is (2²)³ equal to when expressed as a single power of 2?
Explanation22×3 = 2&sup6; = 64. -
Q10 Hard
What is the value of 163/4?
Explanation163/4 = (&sup4;√16)³ = 2³ = 8. -
Q11 Medium
What is the value of √8 × √2?
Explanation√16 = 4. -
Q12 Hard
Rationalize the denominator: 1/(√5 − 2) = ?
ExplanationMultiply by (√5 + 2)/(√5 + 2): denominator = 1. -
Q13 Hard
What is the value of 272/3?
Explanation272/3 = (³√27)² = 3² = 9. -
Q14 Easy
If x1/2 = 5, what is the value of x?
ExplanationSquare both sides: x = 25.
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DS1 Easy
What is √72 when written in its simplest surd form?
Statement I√72 = 6√2.
Statement II72 = 36 × 2.
ExplanationEach alone gives 6√2. -
DS2 Medium
Given that 2m = 8 and 2n = 4, what is the value of 2m+n?
Statement I2m = 8.
Statement II2n = 4.
ExplanationNeed both: m = 3, n = 2, so 25 = 32.
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DS3 Medium
If x2/3 = 4, what is the value of x?
Statement Ix is positive.
Statement IIx1/3 = 2.
ExplanationEach alone: x1/3 = 2 gives x = 8. -
DS4 Hard
When a = 5, what is 1/(√a − 1) equal to after rationalizing the denominator using the conjugate surd?
Statement Ia = 5.
Statement IIThe conjugate is √a + 1.
ExplanationStatement I alone: result is √5 + 1.