👁 Preview — Study, Practice and Revise are open; mock tests and the rest of the syllabus unlock on subscription. Unlock all · ₹4,999
← Back to Number System
Study mode

Surds and Indices

Study Practice Revise 🔒 Test

Introduction to Surds and Indices

In the study of numbers, two important concepts often appear in quantitative aptitude: surds and indices. Understanding these helps us simplify complex expressions and solve a variety of problems efficiently.

Surds are irrational roots-numbers that cannot be expressed as exact fractions, such as \(\sqrt{2}\) or \(\sqrt[3]{5}\). They often appear when dealing with roots that do not simplify to whole numbers.

Indices, also known as exponents or powers, tell us how many times a number (called the base) is multiplied by itself. For example, \(2^3 = 2 \times 2 \times 2 = 8\).

Both surds and indices are fundamental in simplifying expressions, solving equations, and appear frequently in competitive exams. Moreover, they have practical applications in fields like engineering, physics, and finance.

Surds: Definition and Properties

A surd is an irrational root that cannot be simplified to remove the root sign completely. For example, \(\sqrt{2}\) is a surd because it cannot be simplified into a rational number, but \(\sqrt{4} = 2\) is not a surd since it simplifies to a rational number.

Common surds include square roots and cube roots of numbers that are not perfect squares or cubes.

Key properties of surds include:

  • Product Property: \(\sqrt{a} \times \sqrt{b} = \sqrt{ab}\), where \(a, b > 0\).
  • Quotient Property: \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\), where \(a, b > 0\).
  • Like Surds: Surds with the same radicand (number inside the root) can be added or subtracted like algebraic terms.

One important technique is rationalizing the denominator, which means eliminating surds from the denominator of a fraction to simplify calculations and present answers in a standard form.

Step 1: Start with \(\sqrt{50}\) Step 2: Factor 50 as \(25 \times 2\) \(\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}\)

Indices: Laws and Applications

Indices or exponents express repeated multiplication of a base number. For example, \(a^3 = a \times a \times a\).

There are several important laws of indices that help simplify expressions:

Law Formula Example
Product Rule \(a^m \times a^n = a^{m+n}\) \(2^3 \times 2^4 = 2^{3+4} = 2^7 = 128\)
Quotient Rule \(\frac{a^m}{a^n} = a^{m-n}\) \(\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625\)
Power of a Power \((a^m)^n = a^{m \times n}\) \((3^2)^3 = 3^{2 \times 3} = 3^6 = 729\)
Zero Exponent \(a^0 = 1, \quad a eq 0\) \(7^0 = 1\)
Negative Exponent \(a^{-n} = \frac{1}{a^n}\) \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)
Fractional Exponent \(a^{\frac{m}{n}} = \sqrt[n]{a^m}\) \(8^{\frac{2}{3}} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4\)

Rationalization Techniques

Rationalizing the denominator means rewriting a fraction so that the denominator is a rational number (no surds). This is important because it makes calculations easier and answers neater.

There are two common cases:

  1. Single Surd in Denominator: For example, \(\frac{1}{\sqrt{a}}\). Multiply numerator and denominator by \(\sqrt{a}\) to get \(\frac{\sqrt{a}}{a}\).
  2. Binomial Surds in Denominator: For example, \(\frac{1}{\sqrt{a} + \sqrt{b}}\). Multiply numerator and denominator by the conjugate \(\sqrt{a} - \sqrt{b}\) to eliminate surds.
graph TD    A[Start with denominator containing surds] --> B{Is denominator a single surd?}    B -- Yes --> C[Multiply numerator and denominator by the same surd]    B -- No --> D{Is denominator a binomial surd?}    D -- Yes --> E[Multiply numerator and denominator by conjugate]    D -- No --> F[No rationalization needed]    C --> G[Denominator becomes rational]    E --> G    G --> H[End]    F --> H

Worked Examples

Example 1: Simplify \(\sqrt{72} + 3\sqrt{8}\) Easy
Simplify the expression \(\sqrt{72} + 3\sqrt{8}\).

Step 1: Simplify each surd by factoring out perfect squares.

\(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}\)

\(\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}\)

Step 2: Substitute back into the expression:

\(6\sqrt{2} + 3 \times 2\sqrt{2} = 6\sqrt{2} + 6\sqrt{2} = 12\sqrt{2}\)

Answer: \(12\sqrt{2}\)

Example 2: Evaluate \((2^3 \times 2^{-5}) \div 2^{-2}\) Medium
Calculate the value of \((2^3 \times 2^{-5}) \div 2^{-2}\).

Step 1: Apply the product rule of indices for multiplication:

\(2^3 \times 2^{-5} = 2^{3 + (-5)} = 2^{-2}\)

Step 2: Now divide by \(2^{-2}\) using the quotient rule:

\(\frac{2^{-2}}{2^{-2}} = 2^{-2 - (-2)} = 2^0\)

Step 3: Recall that any non-zero number raised to zero is 1:

\(2^0 = 1\)

Answer: \(1\)

Example 3: Rationalize the denominator of \(\frac{5}{\sqrt{3} + \sqrt{2}}\) Medium
Rationalize the denominator of \(\frac{5}{\sqrt{3} + \sqrt{2}}\).

Step 1: Identify the conjugate of the denominator \(\sqrt{3} + \sqrt{2}\), which is \(\sqrt{3} - \sqrt{2}\).

Step 2: Multiply numerator and denominator by the conjugate:

\(\frac{5}{\sqrt{3} + \sqrt{2}} \times \frac{\sqrt{3} - \sqrt{2}}{\sqrt{3} - \sqrt{2}} = \frac{5(\sqrt{3} - \sqrt{2})}{(\sqrt{3} + \sqrt{2})(\sqrt{3} - \sqrt{2})}\)

Step 3: Simplify the denominator using difference of squares:

\((\sqrt{3})^2 - (\sqrt{2})^2 = 3 - 2 = 1\)

Step 4: Therefore, the expression becomes:

\(5(\sqrt{3} - \sqrt{2}) = 5\sqrt{3} - 5\sqrt{2}\)

Answer: \(\frac{5\sqrt{3} - 5\sqrt{2}}{1} = 5\sqrt{3} - 5\sqrt{2}\)

Example 4: Simplify \((27)^{2/3}\) Easy
Simplify \((27)^{2/3}\).

Step 1: Express the fractional exponent as a root and power:

\((27)^{2/3} = \left(\sqrt[3]{27}\right)^2\)

Step 2: Calculate the cube root of 27:

\(\sqrt[3]{27} = 3\) because \(3^3 = 27\)

Step 3: Now square the result:

\(3^2 = 9\)

Answer: \(9\)

Example 5: If the cost of 1 kg rice is Rs.50, find the cost of \(\sqrt{2}\) kg rice Easy
The price of 1 kg rice is Rs.50. Calculate the cost of \(\sqrt{2}\) kg rice.

Step 1: Cost per kg = Rs.50

Step 2: Cost of \(\sqrt{2}\) kg = \(50 \times \sqrt{2}\)

Step 3: Approximate \(\sqrt{2} \approx 1.414\)

Step 4: Multiply:

\(50 \times 1.414 = 70.7\)

Answer: The cost of \(\sqrt{2}\) kg rice is approximately Rs.70.70

Formula Bank

Product Rule of Indices
\[ a^m \times a^n = a^{m+n} \]
where: \(a\) = base, \(m,n\) = exponents
Quotient Rule of Indices
\[ \frac{a^m}{a^n} = a^{m-n} \]
where: \(a\) = base, \(m,n\) = exponents
Power of a Power
\[ (a^m)^n = a^{m \times n} \]
where: \(a\) = base, \(m,n\) = exponents
Zero Exponent Rule
\[ a^0 = 1, \quad a eq 0 \]
where: \(a\) = base
Negative Exponent Rule
\[ a^{-n} = \frac{1}{a^n} \]
where: \(a\) = base, \(n\) = positive exponent
Fractional Exponent Rule
\[ a^{\frac{m}{n}} = \sqrt[n]{a^m} \]
where: \(a\) = base, \(m\) = power, \(n\) = root
Simplification of Surds
\[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \]
where: \(a,b\) = positive real numbers
Rationalization of Denominator (Single Surd)
\[ \frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a} \]
where: \(a\) = positive real number
Rationalization of Denominator (Binomial Surds)
\[ \frac{1}{\sqrt{a} + \sqrt{b}} = \frac{\sqrt{a} - \sqrt{b}}{a - b} \]
where: \(a,b\) = positive real numbers, \(a eq b\)

Tips & Tricks

Tip: Always simplify surds by factoring out perfect squares before performing operations.

When to use: When simplifying expressions involving surds to make addition or subtraction easier.

Tip: Use the conjugate to rationalize denominators with binomial surds quickly.

When to use: When the denominator is of the form \(\sqrt{a} + \sqrt{b}\) or \(\sqrt{a} - \sqrt{b}\).

Tip: Convert fractional indices to roots to better understand and simplify expressions.

When to use: When dealing with fractional powers to avoid confusion.

Tip: Remember that \(a^0 = 1\) for any non-zero \(a\) to quickly simplify expressions.

When to use: When exponents reduce to zero during simplification.

Tip: For negative exponents, rewrite as reciprocal powers to simplify calculations.

When to use: When expressions contain negative powers.

Common Mistakes to Avoid

❌ Adding surds with different radicands directly (e.g., \(\sqrt{2} + \sqrt{3} = \sqrt{5}\))
✓ Surds can only be added or subtracted if their radicands are the same; otherwise, leave them separate.
Why: Students confuse surds with like terms in algebra, but unlike terms cannot be combined.
❌ Incorrectly applying power rules by adding exponents when bases differ (e.g., \(2^3 \times 3^2 = 2^5\))
✓ Laws of indices apply only when bases are the same; multiply bases separately.
Why: Misunderstanding that exponent rules require identical bases.
❌ Forgetting to rationalize denominators in final answers.
✓ Always rationalize denominators to conform to standard answer formats in exams.
Why: Lack of attention to detail and exam conventions.
❌ Misinterpreting fractional exponents as multiplication rather than roots.
✓ Understand that \(a^{m/n}\) means the \(n\)th root of \(a\) raised to the \(m\)th power.
Why: Confusion between fractional exponents and simple multiplication.
❌ Ignoring negative signs when rationalizing denominators with conjugates.
✓ Carefully apply conjugate multiplication and keep track of signs.
Why: Rushing through steps leads to sign errors.
Curated videos per subtopic
Top YouTube explainers, AI-ranked for your exam and language. Unlocks with subscription.
Unlock

Try Practice next.

Progress tracking is paywalled — subscribe to mark subtopics as understood and save your streak.

Go to practice →
Ask a doubt
Surds and Indices · 10 free messages
Ask me anything about this subtopic. You have 10 free messages this session — chat history isn't saved in preview.