GCSE Maths Practice: powers-and-roots
This Higher-tier question explores fractional indices and the power-of-a-power rule. It reinforces how roots and powers interact, an essential skill for simplifying algebraic expressions and exponential problems.
Multiply the exponents directly to simplify. Fractional indices combine roots and powers into one expression, so \((a^{1/2})^3\) becomes \(a^{3/2}\).
Understanding Fractional Powers and the Power-of-a-Power Rule
Fractional powers allow us to represent roots and powers together in one compact form. The rule \(a^{m/n} = \sqrt[n]{a^m}\) connects both ideas: the denominator tells us which root to take, and the numerator shows how many times the number is multiplied by itself. Expressions such as \((a^{1/2})^3\) test your understanding of how to combine these concepts using index laws.
The Power-of-a-Power Rule
The power-of-a-power rule states that \((a^m)^n = a^{m \times n}\). This rule simplifies nested exponents by multiplying the indices. It works with all positive and negative powers, integers, fractions, and decimals. Applying it ensures that multiple exponent operations are expressed as a single power, making calculations and algebraic manipulation easier.
Step-by-Step Method
- Start with the given expression in the form \((a^p)^q\).
- Multiply the exponents: \(a^{p \times q}\).
- Rewrite the result in root form if needed using \(a^{m/n} = \sqrt[n]{a^m}\).
- Calculate or simplify symbolically depending on the base.
This method avoids the need for two separate operations, as it combines the power and root in a single step using fractional indices.
Worked Examples (Different Numbers)
- \((4^{1/2})^3 = 4^{3/2} = (\sqrt{4})^3 = 8\)
- \((16^{1/2})^3 = 16^{3/2} = (\sqrt{16})^3 = 64\)
- \((25^{1/2})^3 = 25^{3/2} = (\sqrt{25})^3 = 125\)
Notice that in each example, the square root is taken first (since the denominator is 2), followed by cubing the result (the numerator). Alternatively, you could multiply the indices directly first to get \(a^{3/2}\) and then evaluate.
Common Mistakes
- Reversing the fraction: \(a^{1/2}\) means square root, not one divided by two.
- Adding exponents instead of multiplying when applying the power-of-a-power rule.
- Calculating in the wrong order when using roots and powers separately — remember, the order does not matter for positive numbers, but following index rules is safer.
- Rounding too early when the result involves decimals, leading to inaccuracies.
Real-Life Applications
Fractional exponents are used in science, engineering, and finance to represent proportional growth or decay. For example, in physics, the relationship between force, mass, and energy often involves square roots and cubes of quantities. In finance, compound interest formulas for fractional time periods depend on powers like \((1 + r)^{t/12}\), which are fractional exponents representing monthly growth rates. In computing, algorithms for data scaling or normalisation often rely on power transformations that use roots and fractional powers.
Quick FAQ
- Q1: What does the numerator and denominator in \(a^{m/n}\) represent?
A1: The numerator is the power, and the denominator is the root. - Q2: Does the order of root and power matter?
A2: For positive numbers, no — both orders give the same result. However, applying powers first is often simpler algebraically. - Q3: How do I know whether to multiply or divide exponents?
A3: When one power is raised to another, multiply them; when two powers of the same base are multiplied together, add them.
Study Tip
Practise rewriting fractional powers both ways — as roots and as powers — to develop flexibility. For example, \(a^{3/2}\) can be viewed as either \((\sqrt{a})^3\) or \(\sqrt{a^3}\). When simplifying complex expressions, always use the power-of-a-power rule to combine exponents efficiently. Mastering fractional indices is essential preparation for algebraic manipulation, surds, and logarithms in higher-level mathematics.