Surds and Indices Practice Questions with Detailed Solutions – Level 3

Surds and Indices Level 3 Practice Quiz: Radicals and Exponents

This Level 3 surds and indices set builds skill with radicals and exponents: rationalising awkward denominators, simplifying nested roots and solving exponential equations. Aptitude360 offers it as advanced Number System practice that also covers exact square and cube roots and the comparison of surds. Every question has a detailed solution.

Topics covered in this surds and indices set

• Rationalising denominators with several surd terms and with cube roots
• Simplifying nested radicals into sums or differences of simple surds
• Infinite nested radicals: repeating square-root nests can lead to quadratic equations, while cube-root nests can lead to cubic equations
• Exponential equations that reduce to quadratics in a single power
• Comparing and ordering surds of different orders
• Exact square roots and cube roots of large numbers and decimals
• Algebraic values built from reciprocal pairs of conjugate surds

Key formulas and rules for surds and indices

• Product rule: \( a^{m}\times a^{n}=a^{m+n} \)
• Quotient rule, for non-zero a: \( a^{m}\div a^{n}=a^{m-n} \)
• Power of a power: \( (a^{m})^{n}=a^{mn} \)
• Root as a power, for positive a: \( \sqrt[n]{a}=a^{\dfrac{1}{n}} \)
• Rationalising, for positive a and b with a not equal to b: \( \dfrac{1}{\sqrt{a}+\sqrt{b}}=\dfrac{\sqrt{a}-\sqrt{b}}{a-b} \)
• Nested radical, for positive a and b: \( \sqrt{a+b+2\sqrt{ab}}=\sqrt{a}+\sqrt{b} \)

Revision tips for advanced surds and indices

• To compare positive surds of different orders, raise them all to a common power such as the LCM of their root orders
• For a nested radical, look for two numbers whose sum and product match the expression inside
• Replace a repeated power with a new variable when an exponential equation looks quadratic
• For exact square and cube roots, estimate likely candidates from the size and last digit of the number, then confirm by squaring or cubing

Who should take this Level 3 set

It is meant for aspirants preparing quantitative aptitude who already know the laws of indices and basic rationalising, and want challenging practice.