Structure and calculation: calculate with roots, and with integer indices

The idea

This row asks you to calculate, not merely recognise, with two linked ideas: roots and integer indices. A root reverses a power, so √49 = 7 because 7² = 49, and ∛64 = 4 because 4³ = 64. An integer index tells you how many times the base multiplies itself: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Negative integer indices give reciprocals: 3⁻² = 1/3² = 1/9. The index laws let you combine powers with the same base: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, and (aᵐ)ⁿ = aᵐⁿ. Any non-zero base to the power zero equals 1, so 7⁰ = 1. In a multiple-choice question, work out the value or simplified form yourself before comparing options, and check signs and reciprocals carefully.

What earns the marks

• Evaluates square roots of perfect squares, for example √81 = 9, and recognises that a negative number has no real square root.
• Evaluates cube roots including negative radicands, for example ∛125 = 5 and ∛(−27) = −3.
• Evaluates positive integer powers, for example 3⁴ = 81, and applies the zero index rule a⁰ = 1 for non-zero a.
• Interprets negative integer indices as reciprocals, for example 5⁻² = 1/25, and evaluates them correctly.
• Applies index laws for multiplication, division and brackets with the same base, for example 4⁵ × 4⁻² = 4³ = 64.
• Selects the single correct option after independent calculation rather than by matching surface features of the options.

Common mistakes and how to put them right

• Treating a negative index as making the answer negative.

  Put it right: A negative index means a reciprocal, so 2⁻³ = 1/2³ = 1/8.

• Confusing the root with division by the index.

  Put it right: √36 asks which number squares to 36, so √36 = 6.

• Adding the bases instead of the indices when multiplying powers.

  Put it right: The base stays the same and the indices add, so 2³ × 2² = 2⁵ = 32.

• Assuming every number has a real square root.

  Put it right: Squaring any real number gives a non-negative result, so √(−16) has no real value, though ∛(−64) = −4 is valid.

Exam tips

• Work out the exact value or simplified power before looking at the options, then match your result to the list.
• Rewrite negative indices as reciprocals first, for example 10⁻³ = 1/1000, so sign errors are easier to spot.
• Check index-law answers by expanding small cases, such as confirming 2³ × 2² = 8 × 4 = 32 = 2⁵.
• For roots, ask which number multiplied by itself the required number of times gives the radicand, and note whether the radicand is negative.