B: Algebra and functions: Understand and use the laws of indices for all rational exponents.

The idea

This topic extends the familiar rules for integer powers to any rational exponent. You need to know that a^(1/n) is the nth root of a, and a^(m/n) is the nth root of a raised to the power m, or equivalently (a^(1/n))^m. The three core laws are: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m-n), and (a^m)^n = a^(mn). These hold for all rational m and n, provided the base a is positive when roots are involved. For example, 8^(2/3) = (8^(1/3))^2 = 2^2 = 4. You also need to handle negative and zero exponents: a^0 = 1 (a ≠ 0) and a^(-n) = 1/a^n. When simplifying expressions, convert roots to fractional powers, combine using the laws, and leave answers in the form requested. This skill underpins later work on exponential functions, logarithms, and calculus.

What earns the marks

• Correctly interpret a^(m/n) as the nth root of a raised to the power m, or as (a^(1/n))^m, and evaluate numerical examples such as 27^(2/3) = 9.
• Apply the multiplication law a^m × a^n = a^(m+n) to simplify expressions with rational exponents, including cases with different bases that cannot be combined.
• Apply the division law a^m ÷ a^n = a^(m-n) and the power law (a^m)^n = a^(mn) correctly, including negative and fractional exponents.
• Use the zero and negative exponent rules: a^0 = 1 and a^(-n) = 1/a^n, and combine these with other laws to simplify expressions.
• Convert between radical and exponent notation, e.g. √x = x^(1/2) and 1/∛x = x^(-1/3), to simplify or evaluate expressions.

Common mistakes and how to put them right

• Assuming (a + b)^n = a^n + b^n, e.g. (4 + 9)^(1/2) = 2 + 3 = 5 instead of √13.

  Put it right: The laws of indices apply to products and quotients, not sums; evaluate the sum first or leave as a single term.

• Mishandling negative exponents in fractions, e.g. simplifying 2x^(-1) as 1/(2x) instead of 2/x.

  Put it right: The negative exponent applies only to the variable, so 2x^(-1) = 2 × (1/x) = 2/x.

• Confusing the addition law with multiplication, e.g. writing 2^3 × 2^4 = 2^12 instead of 2^7.

  Put it right: When multiplying powers with the same base, add the exponents; when raising a power to a power, multiply them.

• Evaluating a^(m/n) by taking the power first and then the root, which can create large intermediate numbers, e.g. 16^(3/4) as (16^3)^(1/4) = 4096^(1/4) = 8.

  Put it right: This is not wrong, but taking the root first, (16^(1/4))^3 = 2^3 = 8, is usually safer; either order gives the same value.

Exam tips

• Always show intermediate steps when simplifying expressions with rational exponents, especially when converting between radical and exponent form, as method marks may be available.
• Check whether the question asks for an exact value or a simplified expression; leave answers in index form if instructed, or evaluate fully if a numerical answer is required.
• When evaluating expressions like 16^(3/4), take the root first (fourth root of 16 is 2) then raise to the power 3 to get 8, as this reduces arithmetic errors.
• Remember that the laws of indices apply only when the bases are the same; if bases differ, look for ways to express them as powers of a common base, e.g. 4^x = 2^(2x).