Nalanda College, Colombo 10 Grade 11 Maths Worksheet: 2-3 Indices and Logarithms I, II

Evaluate a quotient of powers of 2

Answer: 1/4

💡 What it's asking: All the numbers are powers of 2, so we can use the index rules: add indices when multiplying, subtract when dividing.

1. 1On top, 2³ × 2⁻³: same base, so add the indices: 3 + (−3) = 0.
2. 2Any number to the power 0 is 1, so the top is 1.
3. 3Now divide by 2² = 4: 1 ÷ 4 = 1/4.

✅ In ordinary numbers: 8 × 1/8 = 1, and 1 ÷ 4 = 0.25 = 1/4.

Rewrite a root of a negative power with positive index

Answer: x^(3/2)/2

💡 What it's asking: We must write the expression with only positive indices. A square root is the same as the power 1/2, and a negative index means 'flip to the other side of the fraction'.

1. 1Write the square root as a power: √(x⁻³) = x^(−3/2).
2. 2So the expression is 1 ÷ (2 × x^(−3/2)).
3. 3A negative power in the denominator moves up to the numerator and becomes positive: x^(3/2).
4. 4The 2 stays below: answer x^(3/2)/2.

✅ Try x = 4: √(4⁻³) = √(1/64) = 1/8, so 1/(2 × 1/8) = 4; and 4^(3/2)/2 = 8/2 = 4. They match.

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