Number - Structure and calculation: calculate with roots, and with integer indices (Foundation tier)

The idea

The index laws let you handle powers without writing them out in full. Multiplying powers of the same base adds the indices, so 2³ × 2⁴ = 2⁷ = 128. Dividing them subtracts the indices, so 5⁶ ÷ 5⁴ = 5² = 25. Raising a power to another power multiplies the indices, so (3²)³ = 3⁶ = 729. Any non-zero number raised to the power zero is 1, and for a non-zero base a negative index means the reciprocal of the matching positive power, so 2⁻³ = 1/2³ = 1/8. Roots sit alongside these: when a is zero or greater, √a × √a = a, and √36 × √4 = 6 × 2 = 12, which is also √144. The same idea extends to cube roots and higher roots: the cube root of 27 is 3 because 3³ = 27, written ∛27 = 3, and ∛8 × ∛8 × ∛8 = 8. Read whether the answer is wanted as an ordinary number or as a single power of one base, because the two look very different written down.

What earns the marks

• The correct index law used, adding the indices for a product and subtracting them for a quotient.
• A correct intermediate power such as 2⁷, even where it is not then evaluated.
• Evaluating that power to a number where the question wants a value.
• Rewriting a negative index as a fraction, or a fraction as a negative index, whichever direction the question asks for.
• A correct root value, including cube roots such as ∛27 = 3, not only square roots.

Common mistakes and how to put them right

• Multiplying the indices when the powers are multiplied, so writing 2³ × 2⁴ as 2¹².

  Put it right: The indices are added, giving 2⁷.

• Reading a negative index as a negative answer, so giving 2⁻³ as −8.

  Put it right: The correct value is 1/8.

• Combining indices when the bases are different, so turning 2³ × 3² into a single power.

  Put it right: Work each power out separately instead.

• Halving instead of rooting, so giving the square root of 36 as 18.

  Put it right: The correct value is 6.

• Confusing a cube root with a square root, so giving ∛27 as 9.

  Put it right: The correct value is 3 because 3³ = 27, whereas √27 is not 9 (9 is (∛27)² or 27 ÷ 3).

Exam tips

• Check the bases match before you add or subtract any indices; if they do not, work each power out separately.
• Where the answer must be a single power, leave it as a power and do not evaluate it.
• Turn a negative index into a fraction as your first move, because the arithmetic is easier from there.
• For a cube root, ask what number multiplied by itself three times gives the value, so ∛64 = 4 because 4 × 4 × 4 = 64.