GCSE Maths Practice: powers-and-roots

Understanding Fractional Indices in Exact Form

Fractional indices express both powers and roots within a single operation. At Higher GCSE level, you must know how to convert between root notation and index notation fluently. The key rule is \(a^{m/n} = \sqrt[n]{a^m}\), which means that the denominator gives the root and the numerator gives the power. This allows roots to be written as powers of fractions and makes it easier to apply the laws of indices.

The Link Between Roots and Powers

Consider how the square root and cube root work. The square root is the number which, when squared, returns the original value; the cube root does the same for cubes. When these operations are written as powers, they become \(a^{1/2}\) and \(a^{1/3}\) respectively. Extending this logic, \(a^{3/2}\) means taking the square root (denominator 2) and then cubing (numerator 3), or vice versa — both give the same result for positive numbers.

Step-by-Step Method

1. Identify the fractional exponent. The denominator represents the root; the numerator represents the power.
2. Convert to root form using \(a^{m/n} = (\sqrt[n]{a})^m\).
3. Evaluate the root first if it simplifies neatly, then raise the result to the power.
4. If the result cannot be simplified to a whole number, express it as a surd.

Worked Examples (Different Numbers)

• \(9^{3/2} = (\sqrt{9})^3 = 3^3 = 27\)
• \(4^{3/2} = (\sqrt{4})^3 = 2^3 = 8\)
• \(11^{3/2} = (\sqrt{11})^3 = 11\sqrt{11}\)
• \(7^{3/2} = (\sqrt{7})^3 = 7\sqrt{7}\)

These examples show how a fractional index combines two steps in one expression — taking a root and applying a power.

Common Mistakes

• Reversing numerator and denominator: \(a^{2/3}\) is not the same as \(a^{3/2}\).
• Thinking the result must always be a whole number — many fractional indices produce surds.
• Applying powers and roots in the wrong order or missing brackets when substituting into a calculator.
• Forgetting that fractional indices follow the same index laws: when multiplying powers with the same base, add the exponents; when raising a power to another power, multiply them.

Real-Life Applications

Fractional indices appear in physics, engineering, and statistics wherever quantities grow or shrink according to non-integer powers. For example, in mechanics, equations describing energy often involve square roots of squared quantities, effectively applying a power of one-half. In finance, growth rates that compound continuously over time use fractional exponents to represent time fractions. Understanding this concept helps translate formulas between root and power notation.

Quick FAQ

• Q1: What does the numerator and denominator of a fractional exponent represent?
  A1: The numerator represents the power; the denominator represents the root.
• Q2: Does the order of taking the power and root matter?
  A2: For positive numbers, no. Both approaches give the same result because the operations commute.
• Q3: Can fractional exponents give exact results?
  A3: Yes. When the base is a perfect square, cube, or higher power, the result can be simplified exactly without decimals.

Study Tip

When simplifying fractional powers, first convert them into root form to see whether the root gives an integer or a surd. Keep results in exact form (like \(7\sqrt{7}\)) unless the question specifically asks for a decimal approximation. This helps develop accuracy and strengthens your understanding of surds and indices — a core skill for algebra, exponential equations, and A-level preparation.