Surds and Indices: Laws, Rules, Examples and Questions

Surds and indices are important topics in algebra and aptitude. They are used to simplify powers, roots, irrational expressions, square roots, cube roots, equations, and many competitive exam questions. If you understand the laws of indices and the laws of surds, long-looking expressions become much easier.

Indices are powers or exponents. For example, in 2⁵, the number 2 is the base and 5 is the index. It means 2 is multiplied by itself five times. Surds are root expressions that cannot be simplified into rational numbers. For example, √2, √3, and √5 are surds because their exact values cannot be written as terminating or recurring decimals.

This article explains surds and indices in simple language. The existing formulas in this post are kept as they are, and new explanations, rules, examples, tables, rationalising methods, practice questions, and FAQs are added so that the topic becomes complete for school learning and aptitude preparation.

What Are Indices?

An index is a small number written above and to the right of a base. It tells how many times the base is multiplied by itself. Indices are also called exponents or powers.

Example: 3⁴ = 3 × 3 × 3 × 3 = 81.

Here 3 is the base and 4 is the index. The index 4 tells us to multiply 3 four times.

Example: a⁵ means a × a × a × a × a.

What Are Surds?

A surd is an irrational root of a rational number. In simple words, a surd is a root value that cannot be written exactly as a whole number or a simple fraction.

Examples of surds are √2, √3, √5, √7, and √11.

Examples that are not surds are √4, √9, √16, and √25 because they can be simplified as 2, 3, 4, and 5.

So, √8 is a surd because it cannot become a rational number. But it can be simplified as 2√2.

Basic Difference Between Surds and Indices

Point	Indices	Surds
Meaning	Powers or exponents	Irrational roots
Example	23, x5	√2, √5, 3√7
Main idea	Repeated multiplication	Root form
Used in	Powers, algebra, equations	Roots, exact values, simplification

Existing Laws of Indices and Surds

Law of indices
1.
2.
3.
4.
5.
6.

laws of surds

1.
2.
3.
4.
5.
6.

Explanation of Laws of Indices

Product Law

When powers with the same base are multiplied, the indices are added.

a^m × aⁿ = a^m+n

Example: 2³ × 2⁴ = 2⁷ = 128.

Quotient Law

When powers with the same base are divided, the indices are subtracted.

a^m / aⁿ = a^m-n, where a is not equal to 0.

Example: 5⁶ / 5² = 5⁴ = 625.

Power of a Power Law

When a power is raised to another power, multiply the indices.

(a^m)ⁿ = a^mn

Example: (3²)⁴ = 3⁸.

Power of a Product Law

When a product is raised to a power, the power applies to each factor.

(ab)ⁿ = aⁿbⁿ

Example: (2x)³ = 8x³.

Power of a Fraction Law

When a fraction is raised to a power, the numerator and denominator both get the power.

(a/b)ⁿ = aⁿ/bⁿ

Example: (2/3)³ = 8/27.

Zero Index Law

Any non-zero number raised to the power 0 is 1.

a⁰ = 1, where a is not equal to 0.

Example: 11⁰ = 1.

Negative Index Law

A negative index means reciprocal. It does not mean the answer must be negative.

a^-n = 1/aⁿ

Example: 2^-3 = 1/8.

Fractional Index Law

A fractional index represents a root.

a^1/2 = √a and a^1/3 = ∛a.

Example: 25^1/2 = 5.

Example: 27^1/3 = 3.

Explanation of Laws of Surds

Surds follow rules that help us simplify root expressions. These rules are very useful in algebra, geometry, and competitive exams.

Product Rule of Surds

√a × √b = √(ab), where a and b are non-negative.

Example: √2 × √8 = √16 = 4.

Quotient Rule of Surds

√a / √b = √(a/b), where b is not equal to 0.

Example: √18 / √2 = √9 = 3.

Power Form of Surds

A root can be written using a fractional index.

√a = a^1/2.

∛a = a^1/3.

Example: √7 = 7^1/2.

Types of Surds

Pure Surd

A pure surd has no rational coefficient except 1. Examples are √2, √5, and ∛7.

Mixed Surd

A mixed surd has a rational coefficient multiplied by a surd. Examples are 3√2, 5√7, and 2∛3.

Like Surds

Like surds have the same radical part. Examples are 2√3, 5√3, and -7√3. These can be added or subtracted.

Unlike Surds

Unlike surds have different radical parts. Examples are √2 and √3. These cannot be directly added into one surd.

Simplifying Surds

To simplify a surd, look for a perfect-square factor inside the root. Split the number into perfect-square part and remaining part.

Example: Simplify √50.

50 = 25 × 2.

√50 = √25 × √2 = 5√2.

Example: Simplify √72.

72 = 36 × 2.

√72 = √36 × √2 = 6√2.

Example: Simplify √108.

108 = 36 × 3.

√108 = √36 × √3 = 6√3.

Adding and Subtracting Surds

Only like surds can be added or subtracted. Add or subtract the coefficients and keep the surd part same.

Example: 3√5 + 7√5 = 10√5.

Example: 9√2 – 4√2 = 5√2.

Example: 2√3 + 5√7 cannot be combined because √3 and √7 are unlike surds.

Multiplying Surds

To multiply surds, multiply the numbers outside the roots and multiply the numbers inside the roots.

Example: 2√3 × 4√5 = 8√15.

Example: 3√2 × 5√8 = 15√16 = 15 × 4 = 60.

Dividing Surds

To divide surds, divide coefficients and divide the radical parts if possible.

Example: √48 / √3 = √16 = 4.

Example: 6√10 / 2√5 = 3√2.

Rationalising the Denominator

Rationalising means removing the surd from the denominator of a fraction. We do this because answers are usually preferred with rational denominators.

Example: Rationalise 1/√5.

Multiply numerator and denominator by √5.

1/√5 × √5/√5 = √5/5.

Example: Rationalise 3/√7.

3/√7 × √7/√7 = 3√7/7.

Rationalising with Conjugates

If the denominator has two terms such as a + √b or a – √b, multiply numerator and denominator by its conjugate. The conjugate of a + √b is a – √b. The conjugate of a – √b is a + √b.

Example: Rationalise 1/(2 + √3).

Multiply by (2 – √3)/(2 – √3).

1/(2 + √3) = (2 – √3)/(4 – 3) = 2 – √3.

Example: Rationalise 5/(3 – √2).

Multiply by (3 + √2)/(3 + √2).

5/(3 – √2) = 5(3 + √2)/(9 – 2) = 5(3 + √2)/7.

Converting Between Surds and Indices

Surds and indices are connected through fractional powers. This helps in simplifying many algebraic questions.

Surd form	Index form
√a	a1/2
∛a	a1/3
⁴√a	a1/4
√(a3)	a3/2
∛(a2)	a2/3

Example: Write √x as an index.

√x = x^1/2.

Example: Write 16^3/4 in root form.

16^3/4 = (⁴√16)³ = 2³ = 8.

Surds and Indices in Aptitude

In competitive exams, surds and indices are often asked as simplification questions. The main skill is to identify the correct law quickly.

Example 1

Simplify 17^3.5 × 17^x = 17⁸.

Since bases are same, add powers.

3.5 + x = 8.

x = 4.5.

Example 2

Simplify 2^x × 8^1/4 = 2^1/4.

Write 8 as 2³.

8^1/4 = (2³)^1/4 = 2^3/4.

So 2^x × 2^3/4 = 2^1/4.

x + 3/4 = 1/4.

x = -1/2.

Example 3

Simplify √27 + √48.

√27 = √(9 × 3) = 3√3.

√48 = √(16 × 3) = 4√3.

So √27 + √48 = 7√3.

Example 4

Simplify √75 – √12.

√75 = 5√3.

√12 = 2√3.

So √75 – √12 = 3√3.

Example 5

Simplify (√5 + √3)(√5 – √3).

Use (a + b)(a – b) = a² – b².

Answer = 5 – 3 = 2.

Common Mistakes in Surds and Indices

• Adding powers when bases are different.
• Multiplying powers in the product law instead of adding them.
• Adding unlike surds such as √2 + √3 as √5, which is wrong.
• Forgetting to simplify surds before adding or subtracting.
• Forgetting to rationalise the denominator when required.
• Thinking a negative index gives a negative answer. It actually gives a reciprocal.
• Confusing √a + √b with √(a + b). These are not the same.

Practice Questions

1. Simplify 2⁴ × 2⁵.
2. Simplify 7⁸ / 7³.
3. Simplify (x³)⁴.
4. Simplify 5^-2.
5. Simplify 81^1/2.
6. Simplify 27^1/3.
7. Simplify √98.
8. Simplify √45 + √20.
9. Simplify 4√7 – √7.
10. Rationalise 2/√3.
11. Rationalise 1/(5 + √6).
12. Simplify (√11 + √2)(√11 – √2).

Answers to Practice Questions

1. 2⁹ = 512.
2. 7⁵.
3. x¹².
4. 1/25.
5. 9.
6. 3.
7. 7√2.
8. 3√5 + 2√5 = 5√5.
9. 3√7.
10. 2√3/3.
11. (5 – √6)/19.
12. 11 – 2 = 9.

Frequently Asked Questions

What are indices?

Indices are powers or exponents. They show how many times a base is multiplied by itself.

What is a surd?

A surd is an irrational root of a rational number. For example, √2 and √3 are surds.

Is √9 a surd?

No. √9 = 3, and 3 is rational. So √9 is not a surd.

Can unlike surds be added?

Unlike surds cannot be combined into one term. For example, √2 + √3 cannot be simplified as √5.

What are like surds?

Like surds have the same root part. For example, 3√5 and 7√5 are like surds.

What is rationalising the denominator?

Rationalising the denominator means removing the surd from the denominator of a fraction by multiplying with a suitable surd or conjugate.

How are surds and indices connected?

Surds can be written as fractional indices. For example, √a = a^1/2 and ∛a = a^1/3.

Why are surds useful?

Surds give exact values when decimal values are not exact. For example, √2 is more exact than writing 1.414 approximately.

Why Students Find Surds and Indices Difficult

Many students find this topic difficult because surds and indices look different, but they are actually connected. A root can be written as a fractional index, and a fractional index can be changed back into a root. Once this connection is clear, the topic becomes much easier.

Another reason for confusion is that different rules are used in different situations. When powers with the same base are multiplied, we add indices. When a power is raised to another power, we multiply indices. When surds have the same radical part, we add coefficients. When the radical parts are different, we cannot combine them directly.

The best way to learn this topic is to identify the type of expression first. Ask yourself whether the expression contains powers, roots, fractions, products, division, or a surd in the denominator. After that, apply the correct rule step by step.

Important Index Values to Remember

Expression	Value
21	2
22	4
23	8
24	16
25	32
32	9
33	27
42	16
52	25
103	1000

These values appear again and again in simplification questions. Remembering them saves time in exams.

Important Surd Values to Remember

Surd	Approximate value
√2	1.414
√3	1.732
√5	2.236
√7	2.646
√10	3.162

In most school and aptitude questions, exact surd form is preferred. For example, 3√2 is usually better than writing 4.242 approximately, unless the question asks for a decimal value.

More Examples on Indices

Example 1

Simplify 4³ × 4².

The base is same, so add indices.

4³ × 4² = 4⁵ = 1024.

Example 2

Simplify 9⁵ / 9².

The base is same, so subtract indices.

9⁵ / 9² = 9³ = 729.

Example 3

Simplify (2⁴)³.

A power is raised to another power, so multiply indices.

(2⁴)³ = 2¹² = 4096.

Example 4

Simplify (3x²)³.

Apply the power to every factor.

(3x²)³ = 3³x⁶ = 27x⁶.

Example 5

Simplify (16)^3/4.

16^3/4 = (⁴√16)³.

⁴√16 = 2.

So, 16^3/4 = 2³ = 8.

More Examples on Surds

Example 1

Simplify √32.

32 = 16 × 2.

√32 = √16 × √2 = 4√2.

Example 2

Simplify √147.

147 = 49 × 3.

√147 = √49 × √3 = 7√3.

Example 3

Simplify 2√12 + 3√27.

√12 = 2√3, so 2√12 = 4√3.

√27 = 3√3, so 3√27 = 9√3.

Total = 4√3 + 9√3 = 13√3.

Example 4

Simplify √18 × √8.

√18 × √8 = √144 = 12.

Example 5

Simplify (√6 + √2)².

Use (a + b)² = a² + 2ab + b².

(√6 + √2)² = 6 + 2√12 + 2.

√12 = 2√3.

So the answer is 8 + 4√3.

Exam Pattern: Equations With Indices

When both sides of an equation can be written with the same base, compare the indices.

Example: Solve 2^x = 32.

32 = 2⁵.

So 2^x = 2⁵.

Therefore, x = 5.

Example: Solve 3^x+1 = 81.

81 = 3⁴.

So x + 1 = 4.

x = 3.

Example: Solve 5^2x = 625.

625 = 5⁴.

So 2x = 4.

x = 2.

Exam Pattern: Surd Equations

In simple surd equations, isolate the surd first and then square both sides carefully.

Example: Solve √x = 7.

Square both sides.

x = 49.

Example: Solve √(x + 5) = 6.

Square both sides.

x + 5 = 36.

x = 31.

When squaring both sides, always check the answer in the original equation because sometimes extra answers may appear in higher-level surd equations.

Step-by-Step Strategy

1. First simplify powers using laws of indices.
2. Then simplify surds by taking out perfect-square factors.
3. Combine only like surds.
4. Rationalise the denominator if a surd remains below the fraction line.
5. For equations, try to make the same base or isolate the root.
6. Check signs and brackets carefully.

Quick Revision Table

Situation	What to do
Same base multiplication	Add indices
Same base division	Subtract indices
Power raised to power	Multiply indices
Negative index	Take reciprocal
Fractional index	Convert to root
Large surd	Take out perfect-square factor
Surd in denominator	Rationalise denominator
Like surds	Add or subtract coefficients

Word Problems Based on Surds and Indices

Problem 1

The side of a square is √18 cm. Find its area.

Area of a square = side × side.

Area = √18 × √18 = 18 square cm.

This shows why surds can still give simple final answers when multiplied correctly.

Problem 2

The diagonal of a square is 10 cm. Find its side.

For a square, diagonal = side × √2.

So side = 10/√2.

Rationalise the denominator: 10/√2 × √2/√2 = 10√2/2 = 5√2 cm.

Problem 3

A number is written as 2^x. If 2^x × 2³ = 2⁹, find x.

Using the product law, x + 3 = 9.

So x = 6.

Problem 4

Simplify the area expression if length = 3√5 and breadth = 2√10.

Area = length × breadth.

Area = 3√5 × 2√10 = 6√50.

√50 = 5√2.

So area = 30√2 square units.

Shortcuts for Competitive Exams

For indices, always try to make the bases equal. For example, 8 can be written as 2³, 27 can be written as 3³, 81 can be written as 3⁴, and 125 can be written as 5³. This makes comparison of powers easier.

For surds, first check whether the number inside the root has a perfect-square factor. Numbers like 8, 12, 18, 20, 27, 32, 45, 48, 50, 72, 75, 98, 108, and 147 are often used in exams because they can be simplified.

For rationalising, remember two common patterns. If the denominator is a single surd such as √3, multiply numerator and denominator by √3. If the denominator is a binomial such as 2 + √3, multiply by its conjugate 2 – √3.

For addition and subtraction, simplify first. For example, √12 + √27 may look unlike at first, but after simplification it becomes 2√3 + 3√3 = 5√3.

Final Learning Checklist

• Know the laws of indices clearly.
• Know that negative index means reciprocal.
• Know that fractional index means root.
• Know the meaning of pure, mixed, like, and unlike surds.
• Practise simplifying surds by taking out perfect-square factors.
• Practise rationalising simple and binomial denominators.
• Do not combine unlike surds.
• Always check brackets in index questions.

One-Minute Recap

Indices are useful when the same number or variable is multiplied again and again. Surds are useful when a root cannot be written exactly as a rational number. Both topics meet each other through fractional powers. For example, √a and a^1/2 mean the same thing.

In exams, most questions are not difficult if you choose the correct rule. Same base multiplication means add powers. Same base division means subtract powers. A power raised to another power means multiply powers. A surd in the denominator usually needs rationalising. Like surds can be combined, but unlike surds should be left separate unless they can first be simplified.

Conclusion

Surds and indices are closely connected. Indices help us handle powers, while surds help us handle roots and exact irrational values.

To master this topic, learn the laws, simplify step by step, practise rationalising denominators, and avoid combining unlike surds. With these basics, most exam questions become much easier.