Surds, Indices and Simplification Formula Guide: Laws of Indices, Rationalisation and Recurring Decimals for SSC
Rahul Kumar
SSC Exam Expert & Content Editor
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Surds, indices and simplification questions are fast scorers in SSC CGL and CHSL, because they need only a few laws. This surds, indices and simplification formula guide for SSC gives the laws of indices, the rules of surds, the rationalisation and square-root tricks, and the recurring decimal formulas with solved examples.
Laws of Indices
| Law | Formula |
|---|---|
| Same base, multiply | a^m × a^n = a^(m + n) |
| Same base, divide | a^m / a^n = a^(m − n) |
| Power of a power | (a^m)^n = a^(mn) |
| Power of a product | (ab)^n = a^n × b^n |
| Zero power | a^0 = 1 |
| Negative power | a^(−n) = 1 / a^n |
| Fractional power | a^(1/n) = nth root of a |
Rules of Surds
| Rule | Formula |
|---|---|
| Product | √a × √b = √(ab) |
| Quotient | √a / √b = √(a/b) |
| Difference of squares | (√a + √b)(√a − √b) = a − b |
| Square of a sum | (√a + √b)² = a + b + 2√(ab) |
| Rationalising 1/(√a + √b) | (√a − √b)/(a − b) |
| Square root of a surd | √(a + b + 2√(ab)) = √a + √b |
Recurring Decimals
| Form | Value |
|---|---|
| 0.aaa... (one digit repeating) | a / 9 |
| 0.ababab... (two digits repeating) | ab / 99 |
| 0.abbb... (one non-repeating and one repeating digit) | (ab − a) / 90 |
Solved Examples
Example 1. Find 23 × 24.
27 = 128.
Example 2. Find 2−3.
1/2³ = 1/8.
Example 3. Simplify √50.
√(25 × 2) = 5√2.
Example 4. Rationalise 1/(√3 − 1).
(√3 + 1)/(3 − 1) = (√3 + 1)/2.
Example 5. Find the value of (√5 + √3)(√5 − √3).
5 − 3 = 2.
Example 6. Find √(7 + 2√10).
5 + 2 = 7 and 5 × 2 = 10. So the value is √5 + √2.
Example 7. Convert 0.363636... into a fraction.
36/99 = 4/11.
Example 8. Convert 0.1666... into a fraction.
(16 − 1)/90 = 15/90 = 1/6.
BODMAS (VBODMAS)
The order is: Vinculum (bar), Bracket, Of, Division, Multiplication, Addition and Subtraction. Solve inside the brackets in the order ( ), { }, [ ].
Quick Tips
- Break a surd under the root into a perfect square and the rest.
- Use the pair "conjugate" to rationalise a denominator.
- For the root of a + 2√b, find two numbers with sum a and product b.
- Learn the recurring decimal formulas.
Quick Revision Points
- a^m × a^n = a^(m + n).
- a^0 = 1 and a^(−n) = 1/a^n.
- (√a + √b)(√a − √b) = a − b.
- 0.ababab... = ab/99.
- VBODMAS is the order of simplification.
Practice Questions With Answers
- Find 25 × 23.
Answer: 256 - Find 34 ÷ 32.
Answer: 9 - Find (23)2.
Answer: 64 - Find the value of 70.
Answer: 1 - Find 2−2.
Answer: 1/4 - Simplify √72.
Answer: 6√2 - Find (√7 + √2)(√7 − √2).
Answer: 5 - Convert 0.333... into a fraction.
Answer: 1/3 - Rationalise 1/(√2 + 1).
Answer: √2 − 1 - Find √(11 + 2√30).
Answer: √6 + √5
How to Practise This Chapter
Write each formula on a card, with a solved example on the back, and revise the cards every day. Then test yourself with our practice papers and the previous year papers page. See all guides on the Formula Guide page, and also read the Mathematics notes and Quantitative Aptitude notes. The previous guide is Time and Work, Pipes and Cisterns and the next guide is Height and Distance.
Frequently Asked Questions
What is a surd?
A surd is an irrational root, such as √2 or √3.
What is the law of indices for multiplication?
a^m × a^n = a^(m + n).
What is a to the power zero?
It is 1, for any a other than 0.
What is a to the power of minus n?
It is 1 divided by a^n.
How do I rationalise a denominator?
Multiply the numerator and the denominator by the conjugate.
What is the conjugate of √a + √b?
It is √a − √b.
How do I convert 0.ababab to a fraction?
Write ab/99.
What is VBODMAS?
It is the order: Vinculum, Bracket, Of, Division, Multiplication, Addition and Subtraction.
Is this topic asked in SSC exams?
Yes. Surds, indices and simplification are asked in SSC CGL and CHSL.
How much time should I take?
About 30 seconds.
What are common mistakes?
Using the laws of indices on different bases.
How can I learn the formulas?
Write them on cards and solve one example for each.
Previous chapter
Time and Work, Pipes and Cisterns Formula Guide for SSC Exams
Next chapter
Height and Distance Formula Guide: Angle of Elevation, Depression and Two-Angle Questions for SSC
Rahul Kumar
SSC Exam Expert & Content Editor
Rahul is a senior SSC exam strategist with 8+ years of experience helping aspirants crack CGL, CHSL and MTS exams. He writes in-depth notification breakdowns, exam pattern guides and preparation strategies.