CAT 2026 Surds and Indices: Formulas, Shortcuts, Questions & Preparation Tips

CAT 2026 Surds and Indices: Basic Concepts

Surds and indices involve roots, powers and exponent rules, which are essential for simplifying CAT expressions quickly.

A surd is an irrational root expression such as $\sqrt{2}$, $\sqrt{3}$, or $\sqrt[5]{3}$. An index or exponent shows the power of a number, as in $a^n$.

The key connection is:

$a^{\frac{1}{n}}=\sqrt[n]{a}$ and

$a^{\frac{m}{n}}=\sqrt[n]{a^m}$

For example, $16^{\frac{3}{4}}=\left(\sqrt[4]{16}\right)^3=2^3=8$.

CAT 2026 Indices: Important Laws

The following laws form the foundation of most indices questions in CAT.

These rules should be applied only after identifying whether the bases are the same and whether the expression involves multiplication, division, or a power raised to another power.

CAT 2026 Surds: Important Formulas

Surd simplification mainly uses factorisation and standard identities.

Square Root of a Product

$\sqrt{ab}=\sqrt{a}\sqrt{b}$

Example: $\sqrt{72}=\sqrt{36\times2}=6\sqrt{2}$

Square Root of a Quotient

$\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$

Addition and Subtraction of Surds

Only like surds can be combined:

$a\sqrt{x}+b\sqrt{x}=(a+b)\sqrt{x}$

Example: $3\sqrt{5}+7\sqrt{5}=10\sqrt{5}$

Multiplication of Surds

$\sqrt{a}\times\sqrt{b}=\sqrt{ab}$

Example: $\sqrt{6}\times\sqrt{24}=\sqrt{144}=12$

CAT 2026 Rationalisation of Surds

Rationalisation removes a surd from the denominator.

For $\frac{1}{\sqrt{a}}$:

$\frac{1}{\sqrt{a}}\times\frac{\sqrt{a}}{\sqrt{a}}=\frac{\sqrt{a}}{a}$

Rationalising a Binomial Surd

For $\frac{1}{\sqrt{a}+\sqrt{b}}$, multiply by the conjugate $\sqrt{a}-\sqrt{b}$:

$\frac{1}{\sqrt{a}+\sqrt{b}}\times\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}-\sqrt{b}}=\frac{\sqrt{a}-\sqrt{b}}{a-b}$

The identity $(a+b)(a-b)=a^2-b^2$ makes rationalisation faster.

CAT 2026 Surds and Indices Shortcuts

These shortcuts can reduce calculation time in CAT-level Surds and Indices questions.

Shortcut 1: Convert Roots into Indices

Write roots as fractional powers for easier multiplication and division.

$\sqrt[3]{x^2}=x^{\frac{2}{3}}$

Shortcut 2: Look for Perfect Powers

Check whether the base is a perfect square, cube, or higher power.

Example: $81^{\frac{3}{4}}=(3^4)^{\frac{3}{4}}=3^3=27$

Shortcut 3: Use a Common Base

Convert numbers to the same base before applying exponent rules.

Example: $8^{\frac{2}{3}}\times4^{\frac{1}{2}}=(2^3)^{\frac{2}{3}}\times(2^2)^{\frac{1}{2}}=2^2\times2=8$

Shortcut 4: Use Conjugates

For denominators such as $\sqrt{a}+\sqrt{b}$ or $\sqrt{a}-\sqrt{b}$, multiply by the conjugate.

Example: $\frac{1}{\sqrt{3}+\sqrt{5}}=\frac{\sqrt{3}-\sqrt{5}}{3-5}=\frac{\sqrt{5}-\sqrt{3}}{2}$

CAT 2026 Surds and Indices: Important Identities

These identities help simplify expressions involving square roots and powers.

Identity 1: $(a+b)^2=a^2+b^2+2ab$

Identity 2: $(a-b)^2=a^2+b^2-2ab$

Identity 3: $(a+b)(a-b)=a^2-b^2$

Identity 4: $(a+b)^3=a^3+b^3+3ab(a+b)$

Identity 5: $x^2+\frac{1}{x^2}=\left(x+\frac{1}{x}\right)^2-2$

$x^3+\frac{1}{x^3}=\left(x+\frac{1}{x}\right)^3-3\left(x+\frac{1}{x}\right)$

CAT 2026 Surds and Indices Sample Questions

Practise CAT-level Surds and Indices questions to strengthen concept clarity, simplify expressions faster, and improve calculation accuracy.

1. If $x=\sqrt{3}+\sqrt{2}$, then the value of $x^2$ is:

   A. $5+2\sqrt{6}$
   B. $5+\sqrt{6}$
   C. $3+2\sqrt{6}$
   D. $6+2\sqrt{5}$

2. The value of $\sqrt{5+2\sqrt{6}}$ is:

   A. $\sqrt{3}+\sqrt{2}$
   B. $\sqrt{3}-\sqrt{2}$
   C. $\sqrt{5}+\sqrt{2}$
   D. $\sqrt{6}+1$

3. If $x=2^{\frac{1}{3}}+2^{-\frac{1}{3}}$, then the value of $x^3-3x$ is:

   A. $1$
   B. $\frac{5}{2}$
   C. $\frac{3}{2}$
   D. $3$

4. The value of $3^5\times9^{-2}\times27$ is:

   A. $3$
   B. $9$
   C. $27$
   D. $81$

5. If $x+\frac{1}{x}=3$, then the value of $x^2+\frac{1}{x^2}$ is:

   A. $5$
   B. $7$
   C. $9$
   D. $11$

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CAT 2026 Surds and Indices: Important Question Types

Recognising the question type before beginning the calculation can prevent unnecessary expansion and lengthy calculations.

Common Mistakes in CAT Surds and Indices Questions

Surds and indices questions often become difficult because of small algebraic errors rather than complicated concepts.

How to Prepare Surds and Indices for CAT 2026

A structured approach can make this topic significantly more manageable.

1. Learn the Index Laws

Start with multiplication, division, zero, negative and fractional exponents. These rules form the foundation of the topic.

2. Practise Surd Simplification

Solve expressions involving square roots, cube roots and combinations of like surds.

3. Master Rationalisation

Practise both single-term and binomial denominators. Conjugates should become an automatic step when required.

4. Learn Common Perfect Powers

Memorise common squares, cubes and fourth powers. This reduces calculation time when simplifying expressions.

5. Practise Mixed Questions

Once the basic concepts are clear, solve questions combining surds, indices, algebraic identities and equations.

6. Use Timed Practice

After mastering the concepts, solve mixed questions under a time limit. The objective is to identify the shortest valid approach rather than perform lengthy calculations.

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CAT 2026 Surds and Indices Quick Revision Formula Sheet

This formula sheet summarises the essential Surds and Indices rules for quick CAT 2026 revision and practice.

A quick review of these formulas can help candidates simplify expressions faster and avoid common calculation errors.

Revise essential CAT Quant formulas in one place for faster calculations, quick revision and stronger exam preparation.

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CAT 2026 Mock Test

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Conclusion

Surds and Indices become much easier once exponent laws, fractional powers, surd identities and rationalisation techniques are familiar. For CAT 2026 preparation, candidates should focus on recognising patterns quickly rather than relying on lengthy calculations. Regular practice with mixed questions can improve both accuracy and speed in Quantitative Aptitude.