Surds and Indices Notes for SSC Exams

Surds and Indices is a short and scoring chapter of SSC Quantitative Aptitude. One or two questions come from it in most exams, and they are based on a handful of laws. These surds and indices notes for SSC explain every law with examples, the rationalisation method and practice questions.

Laws of Indices

What Is a Surd?

A surd is an irrational root, such as the square root of 2 or the cube root of 5. The square root of 4 is not a surd because it equals 2. Surds follow these rules:

• Square root of a x square root of b = square root of (ab).
• Square root of a / square root of b = square root of (a/b).
• Like surds (same number under the root) can be added or subtracted: 5 root 2 + 3 root 2 = 8 root 2.
• To simplify a surd, take out perfect squares: square root of 50 = 5 root 2.

Rationalisation

To remove a surd from the denominator, multiply the numerator and the denominator by its conjugate. The conjugate of (a + root b) is (a - root b). For example, 1 / (root 5 - root 3) = (root 5 + root 3) / ((root 5) squared - (root 3) squared) = (root 5 + root 3) / 2.

Standard Results

• (root a + root b)(root a - root b) = a - b.
• Square root of (a + b + 2 root ab) = root a + root b.
• If 3 + 2 root 2 is under a root, write it as (1 + root 2) squared.
• Roots can be compared by raising them to the same power (LCM of the root orders).

Comparing Surds and Indices

To compare two roots with different orders, such as the square root of 3 and the cube root of 4, write both as powers of the same order. The LCM of 2 and 3 is 6, so square root of 3 = 3^(3/6) = 27^(1/6) and cube root of 4 = 4^(2/6) = 16^(1/6). Since 27 is more than 16, the square root of 3 is greater. For two surds of the same order, simply compare the numbers under the roots.

Common Mistakes in Indices Questions

• Writing 2^3 x 3^3 as 6^6; the correct value is 6^3.
• Writing (a^m)(b^m) as (ab)^(2m); the correct form is (ab)^m.
• Forgetting that a negative power only inverts the number and does not make it negative.
• Treating 0^0 as 1 without the question stating it.

When a question mixes different bases, first try to write every base as a power of the smallest common base, such as 4 = 2 squared, 8 = 2 cubed, 9 = 3 squared and 27 = 3 cubed. Then use the laws of indices.

Solved Examples

Example 1: Simplify: 2^5 x 2^3 / 2^6
Solution: Add the powers in the numerator: 2^8. Divide: 2^(8 - 6) = 2^2 = 4.

Example 2: Find the value of 27^(2/3).
Solution: 27^(1/3) = 3, so 27^(2/3) = 3 squared = 9.

Example 3: If 3^x = 81, find x.
Solution: 81 = 3^4, so x = 4.

Example 4: Rationalise 1 / (root 5 - root 3).
Solution: Multiply by the conjugate: (root 5 + root 3) / (5 - 3) = (root 5 + root 3) / 2.

Example 5: Simplify: root 50 + root 18
Solution: Root 50 = 5 root 2 and root 18 = 3 root 2. Sum = 8 root 2.

Example 6: Find the value of 16^(-3/4).
Solution: 16^(1/4) = 2, so 16^(3/4) = 8, and 16^(-3/4) = 1/8.

Quick Tips to Solve Faster

• Convert every number into a common base before applying the laws.
• Break surds into perfect square factors to add or subtract them.
• Rationalise using the conjugate whenever there is a surd in the denominator.
• Use (root a + root b)(root a - root b) = a - b to simplify fast.
• Write a negative power as a reciprocal.

Common Mistakes to Avoid

• Adding the powers when dividing the same base.
• Treating (a + b)^n as a^n + b^n.
• Forgetting the negative sign while taking a reciprocal.
• Adding unlike surds, such as root 2 + root 3.
• Not rationalising the denominator when required.

Quick Revision Points

• a^m x a^n = a^(m + n) and a^m / a^n = a^(m - n).
• a^0 = 1 and a^(-n) = 1 / a^n.
• a^(1/n) = nth root of a.
• Conjugate of (a + root b) is (a - root b).
• Only like surds can be added or subtracted.

Practice Questions With Answers

1. Find the value of 5^0 + 2^(-1).
   Answer: 1.5 (1 + 0.5).
2. Find the value of 8^(1/3) x 81^(1/4).
   Answer: 6 (2 x 3).
3. If 2^(x + 1) = 32, find x.
   Answer: 4 (2^5 = 32).
4. Express root 72 in the form a root 2. Find a.
   Answer: 6 (root 36 x root 2).
5. Find the value of (2^3)^2.
   Answer: 64 (2^6).
6. Rationalise 1 / (2 + root 3).
   Answer: 2 - root 3 ((2 - root 3) / (4 - 3)).
7. Find the value of 9^(3/2).
   Answer: 27 (3 cubed).
8. Find the value of the square root of (3 + 2 root 2).
   Answer: 1 + root 2 ((1 + root 2) squared).
9. If 4^x = 1/16, find x.
   Answer: -2 (4^(-2)).
10. Find the value of 7^5 / 7^3.
    Answer: 49 (7^2).

How to Practise This Chapter

1. Learn the formulas in the table until you can write them without looking.
2. Solve the solved examples again on your own, without reading the solution.
3. Attempt the practice questions above with a timer of 10 minutes.
4. Solve the matching SSC surds and indices practice paper to test yourself in exam style.
5. Check the previous year papers to see how SSC frames this topic.

Related Notes and Practice Papers

• Algebra Formulas Notes
• Square Root and Cube Root Notes
• Simplification and BODMAS Notes
• Quadratic Equations Notes
• SSC Surds and Indices Practice Paper

Explore More SSC Subjects

SSC exams test many subjects, so connect this topic with the related subjects below:

• Reasoning Notes
• Statistics Notes
• Indian Economy Notes
• English Notes
• General Awareness Notes
• All SSC subjects and notes

Frequently Asked Questions

What are indices?

Indices (exponents or powers) show how many times a number is multiplied by itself.

What is a surd?

A surd is an irrational root that cannot be simplified into a rational number, such as root 2 or root 7.

What is the zero power rule?

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

How do I rationalise a denominator?

Multiply the numerator and the denominator by the conjugate of the denominator.

Can unlike surds be added?

No. Only like surds, with the same number under the root, can be added or subtracted.

How many questions come from surds and indices in SSC exams?

One or two questions come from this chapter in most SSC Tier 1 papers.

What is a fractional power?

A power such as a^(1/n) means the nth root of a, and a^(m/n) means the nth root of a raised to the power m.

How should I practise surds and indices?

Learn the laws, solve the examples here and attempt the surds and indices practice paper.