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Surds and Indices

Practice MCQs

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Quantitative Aptitude Surds and Indices Competitive Exams

Surds and Indices is an important quantitative aptitude topic based on powers, roots, exponents, rationalization, simplification of radicals, and laws of indices. These questions are commonly asked in competitive exams, aptitude tests, entrance exams, and mathematical reasoning sections.

What are Surds and Indices?

Indices are powers or exponents used to show repeated multiplication. For example, \(2^3\) means \(2 \times 2 \times 2 = 8\).

Surds are irrational roots that cannot be simplified into exact rational numbers. For example, \(\sqrt{2}\), \(\sqrt{3}\), and \(\sqrt{5}\) are surds, while \(\sqrt{4}=2\) is not a surd.

Quick idea: Indices deal with powers. Surds deal with roots that do not simplify completely.
Concept Meaning Example
Index Power or exponent of a number \(2^4=16\)
Base Number being multiplied repeatedly In \(3^2\), base is 3
Exponent Number showing how many times base is multiplied In \(3^2\), exponent is 2
Surd Irrational root \(\sqrt{7}\)
Rationalization Removing surd from denominator \(\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}\)

“In surds and indices, most questions become easy if you apply the correct law at the right step.”

Aptitude Tip
Key points
  • Indices represent repeated multiplication.
  • Same base multiplication means add powers.
  • Same base division means subtract powers.
  • A zero power gives value 1.
  • Surds are irrational roots.
  • Rationalization removes surds from denominators.
indices powers roots surds rationalization

Visual Understanding

These diagrams show how powers, roots, surds, and rationalization work.

Index as Repeated Multiplication
2³ means 2 × 2 × 2 2 × 2 × 2 2³ = 8
\[ a^n = a \times a \times a \times \cdots \times a \]

The exponent tells how many times the base is multiplied by itself.

Surd as Irrational Root
√4 = 2 Not a surd √5 Surd Exact square roots are not surds. Irrational roots are surds.
\[ \sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7} \text{ are surds} \]

A surd cannot be expressed as a simple rational number.

Multiplying Same Bases
a² × a³ = a⁵ Same base: add the powers
\[ a^m \times a^n = a^{m+n} \]

When bases are the same and powers are multiplied, add the exponents.

Rationalization
1 √2 multiply by √2/√2 √2 2 Rationalization removes surd from denominator
\[ \frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2} \]

Rationalization makes the denominator rational.

Important Formulas and Rules

Product Law
\[ a^m \times a^n = a^{m+n} \]

Same base multiplication means add powers.

Quotient Law
\[ \frac{a^m}{a^n}=a^{m-n} \]

Same base division means subtract powers.

Power of Power
\[ (a^m)^n=a^{mn} \]

Multiply the powers.

Zero Power
\[ a^0=1 \]

Any non-zero number raised to zero is 1.

Negative Power
\[ a^{-n}=\frac{1}{a^n} \]

Negative exponent gives reciprocal.

Fractional Power
\[ a^{\frac{1}{n}}=\sqrt[n]{a} \]

Fractional powers represent roots.

Power of Product
\[ (ab)^n=a^n b^n \]

Power applies to both factors.

Power of Quotient
\[ \left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} \]

Power applies to numerator and denominator.

Surd Product
\[ \sqrt{a}\times\sqrt{b}=\sqrt{ab} \]

Product of square roots can be combined.

Surd Quotient
\[ \frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}} \]

Valid when \(b \ne 0\).

Simplifying Surd
\[ \sqrt{ab}=\sqrt{a}\sqrt{b} \]

Split into perfect square and non-square factors.

Conjugate Rule
\[ (a+\sqrt{b})(a-\sqrt{b})=a^2-b \]

Used in rationalization.

Rule: Apply index laws only when the bases are suitable. For surds, first check whether the number contains a perfect square factor.

Common Types of Questions

Index Law Questions

Questions based on multiplication, division, and powers of powers.

  • Same base multiplication
  • Same base division
  • Power of a power
  • Zero and negative powers
Surd Simplification

Simplify square roots by taking out perfect square factors.

  • \(\sqrt{12}=2\sqrt{3}\)
  • \(\sqrt{18}=3\sqrt{2}\)
  • Like surds
  • Unlike surds
Rationalization

Remove the irrational term from the denominator.

  • Single surd denominator
  • Binomial surd denominator
  • Use conjugate
  • Simplify final result
Equation-Based Questions

Use index laws to solve unknown powers or compare expressions.

  • Equating powers
  • Finding unknown exponent
  • Fractional exponents
  • Negative exponents
Exam approach: In indices, convert all terms to the same base when possible. In surds, remove perfect square factors first.
Method Bank
Multiply Same Base

Add the exponents.

\[ 2^3\times2^4=2^7 \]
Divide Same Base

Subtract the exponents.

\[ \frac{5^6}{5^2}=5^4 \]
Simplify Surd

Take out perfect square factors.

\[ \sqrt{50}=\sqrt{25\times2}=5\sqrt{2} \]
Rationalize

Multiply by the same surd.

\[ \frac{3}{\sqrt{5}}=\frac{3\sqrt{5}}{5} \]

Tip: For surds, always look for perfect square factors like 4, 9, 16, 25, 36, 49, and 64.

Surds and Indices Solving Flow
Check Form Apply Law Simplify Final Answer Same base → index laws | Root form → surd rules
First decide whether the expression is power-based or root-based.
\[ \text{Same base} \Rightarrow \text{Use index laws} \]
\[ \text{Square root} \Rightarrow \text{Check perfect square factors} \]

Step-by-Step Solving Method

Step Indices Surds
Step 1 Identify the base and exponent. Identify the number under the root.
Step 2 Check if bases are same or can be made same. Check for perfect square factors.
Step 3 Apply product, quotient, or power law. Simplify the root if possible.
Step 4 Handle zero, negative, or fractional powers. Combine like surds only.
Step 5 Write the answer in simplest power form. Rationalize denominator if required.
Important: Like surds can be added or subtracted, but unlike surds cannot be directly combined.

Solved Examples

Question Method Answer
Simplify: \(2^3 \times 2^4\) Same base, so add powers.
\[ 2^3 \times 2^4 = 2^{3+4}=2^7 \]
\[ 2^7=128 \]
128
Simplify: \(\frac{5^6}{5^2}\) Same base, so subtract powers.
\[ \frac{5^6}{5^2}=5^{6-2}=5^4 \]
\[ 5^4=625 \]
625
Simplify: \((3^2)^4\) Power of a power means multiply powers.
\[ (3^2)^4=3^{2\times4}=3^8 \]
\(3^8\)
Simplify: \(4^{-2}\) Negative power gives reciprocal.
\[ 4^{-2}=\frac{1}{4^2}=\frac{1}{16} \]
\(\frac{1}{16}\)
Simplify: \(\sqrt{50}\) Split into perfect square and remaining factor.
\[ \sqrt{50}=\sqrt{25\times2} \]
\[ \sqrt{50}=5\sqrt{2} \]
\(5\sqrt{2}\)
Simplify: \(3\sqrt{2}+5\sqrt{2}\) These are like surds, so add coefficients.
\[ 3\sqrt{2}+5\sqrt{2}=8\sqrt{2} \]
\(8\sqrt{2}\)
Rationalize: \(\frac{1}{\sqrt{3}}\) Multiply numerator and denominator by \(\sqrt{3}\).
\[ \frac{1}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \]
\(\frac{\sqrt{3}}{3}\)
Rationalize: \(\frac{1}{2+\sqrt{3}}\) Multiply by conjugate \(2-\sqrt{3}\).
\[ \frac{1}{2+\sqrt{3}}\times\frac{2-\sqrt{3}}{2-\sqrt{3}} \]
\[ =\frac{2-\sqrt{3}}{4-3}=2-\sqrt{3} \]
\(2-\sqrt{3}\)

Note: In indices, check the base carefully. In surds, simplify roots before adding, subtracting, or rationalizing.

Common Traps and Shortcuts

Common Traps
  • Multiplying powers instead of adding them for same base multiplication.
  • Adding powers when bases are different.
  • Forgetting that \(a^0=1\).
  • Incorrectly simplifying negative powers.
  • Adding unlike surds directly.
  • Forgetting to rationalize the denominator when required.
Useful Shortcuts
  • Same base multiplication: add powers.
  • Same base division: subtract powers.
  • Power of power: multiply powers.
  • Negative power means reciprocal.
  • Take out perfect square factors from surds.
  • Use conjugate for binomial rationalization.
Exam approach: Convert expressions into simpler base or root form before applying formulas.

Practice

A) Multiple Choice Questions
  1. Simplify: \(3^2 \times 3^3\)
    \(3^5\) \(3^6\) \(6^5\) \(9^3\)
  2. Simplify: \(\frac{2^7}{2^4}\)
    2 4 8 16
  3. Simplify: \(\sqrt{72}\)
    \(6\sqrt{2}\) \(4\sqrt{3}\) \(3\sqrt{8}\) \(2\sqrt{6}\)
  4. \(a^0\), where \(a \ne 0\), is equal to:
    0 1 a \(\frac{1}{a}\)
  5. Rationalize: \(\frac{1}{\sqrt{5}}\)
    \(\sqrt{5}\) \(\frac{\sqrt{5}}{5}\) \(\frac{5}{\sqrt{5}}\) 5
B) Solve the Higher-Order Problems
  1. Simplify: \(5^3 \times 5^2 \div 5\) (Hint: Use laws of indices.)
  2. Simplify: \((2^3)^4\) (Hint: Power of power means multiply powers.)
  3. Simplify: \(\sqrt{108}\) (Hint: \(108=36\times3\).)
  4. Simplify: \(4\sqrt{3}+7\sqrt{3}-2\sqrt{3}\) (Hint: Add and subtract coefficients of like surds.)
  5. Rationalize: \(\frac{2}{3-\sqrt{5}}\) (Hint: Multiply by conjugate \(3+\sqrt{5}\).)
C) Match the Concept with the Correct Meaning
Concept Correct Meaning
Index Power or exponent of a number
Base Number being raised to a power
Surd Irrational root
Rationalization Removing surd from denominator
Conjugate Expression used to rationalize binomial surds
Like Surds Surds having the same irrational part
Aptitude Reminder

Surds and indices questions become easy when you identify the structure first. Same bases need index laws, while roots need surd simplification and rationalization.

Task: Create five questions using product law, quotient law, surd simplification, like surds, and rationalization.

Show Suggested Answers
Multiple Choice
  1. \(3^5\)
    \[ 3^2 \times 3^3 = 3^{2+3}=3^5 \]
  2. 8
    \[ \frac{2^7}{2^4}=2^{7-4}=2^3=8 \]
  3. \(6\sqrt{2}\)
    \[ \sqrt{72}=\sqrt{36\times2}=6\sqrt{2} \]
  4. 1
    \[ a^0=1,\quad a\ne0 \]
  5. \(\frac{\sqrt{5}}{5}\)
    \[ \frac{1}{\sqrt{5}}\times\frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{5} \]
Higher-Order Problems
  1. Simplify:
    \[ 5^3 \times 5^2 \div 5 = \frac{5^{3+2}}{5^1} = 5^{5-1}=5^4 \]
    \[ 5^4=625 \]
    Answer = 625.
  2. Power of a power:
    \[ (2^3)^4=2^{3\times4}=2^{12} \]
    Answer = \(2^{12}\).
  3. Simplify:
    \[ \sqrt{108}=\sqrt{36\times3}=6\sqrt{3} \]
    Answer = \(6\sqrt{3}\).
  4. Like surds:
    \[ 4\sqrt{3}+7\sqrt{3}-2\sqrt{3} = (4+7-2)\sqrt{3} = 9\sqrt{3} \]
    Answer = \(9\sqrt{3}\).
  5. Rationalize using conjugate:
    \[ \frac{2}{3-\sqrt{5}} \times \frac{3+\sqrt{5}}{3+\sqrt{5}} \]
    \[ = \frac{2(3+\sqrt{5})}{9-5} = \frac{2(3+\sqrt{5})}{4} = \frac{3+\sqrt{5}}{2} \]
    Answer = \(\frac{3+\sqrt{5}}{2}\).
Concept Matching
  1. Index → Power or exponent of a number
  2. Base → Number being raised to a power
  3. Surd → Irrational root
  4. Rationalization → Removing surd from denominator
  5. Conjugate → Expression used to rationalize binomial surds
  6. Like Surds → Surds having the same irrational part
Clue Explanation

Indices are simplified using exponent laws. Surds are simplified by removing perfect square factors and rationalizing the denominator when needed.

Exam tips
  • Same base multiplication means add powers.
  • Same base division means subtract powers.
  • Power of power means multiply powers.
  • Negative power means reciprocal.
  • Simplify surds using perfect square factors.
  • Use conjugate for rationalizing binomial surds.
Practice MCQs