Take the number 2 and multiply it by itself ten times. Writing 2 × 2 × 2 and so on gets tiring fast. A small raised number, called an index, does the job in one symbol.
The same page also covers surds, which are roots that never end. This article gives the laws, the shortcuts and five solved examples. Each example says why the method works.
- Multiply, same base: add the indices. Divide, same base: subtract them.
- Power of a power: multiply the indices.
- Zero index: any non-zero number to the power 0 is 1. A negative index flips the number to the other side of a fraction.
- Fractional index: a^(1/n) is the n-th root of a.
- A surd is a root of a rational number that is itself irrational, such as √2.
- To simplify a surd, pull out the largest perfect square. To rationalise, multiply top and bottom by the conjugate.
What are indices and how do they work?
What does an index tell you?
In 5^3, the number 5 is the base and 3 is the index, also called the exponent or power.
It says: multiply the base by itself 3 times. So 5^3 = 5 × 5 × 5 = 125.
Sources: NCERT Class 8 Maths: Exponents and Powers, laws of exponents; Practice question: multiplying powers of 5.
What happens when you multiply or divide powers of one base?
For the same base, add the indices when you multiply and subtract them when you divide.
So 5^12 × 5^5 = 5^17. And 2^15 ÷ 2^7 = 2^8, which is 256.
Sources: NCERT Class 8 Maths: Exponents and Powers, laws of exponents; Practice question: dividing powers of 2; Practice question: multiply then divide powers of 10.
What happens when a power is raised to another power?
Multiply the indices. The rule is (a^m)^n = a^(mn).
So (9^8)^4 = 9^32. First the 8 copies are formed, and then that whole block is repeated 4 times.
Sources: NCERT Class 8 Maths: Exponents and Powers, power of a power; Practice question: nine to the eighth, to the fourth.
What do zero and negative indices mean?
Any non-zero number to the power 0 equals 1. A negative index means one over the positive power.
So 7^0 = 1, and 2^(−3) = 1/2^3 = 1/8.
Sources: NCERT Class 8 Maths: Exponents and Powers, zero and negative exponents.
What is a fractional index?
A fractional index means a root. So a^(1/n) is the n-th root of a.
For example, 27^(1/3) = 3, because 3 × 3 × 3 = 27. And 16^(3/4) = (16^(1/4))^3 = 2^3 = 8.
Sources: NCERT Class 8 Maths: Exponents and Powers, laws of exponents.
What are surds and how do you handle them?
What is a surd?
A surd is a root of a rational number that is irrational. Its decimal never ends and never repeats.
So √2 and ∛5 are surds. But √4 = 2 is not a surd, because it is a whole number.
Sources: NCERT Class 9 Maths: The World of Numbers, irrational numbers; Practice question: definition of a surd.
What is the order of a surd?
The order of a surd is the number of the root. In ⁿ√a, the order is n.
So the 5th root of 34 is a surd of order 5. A square root has order 2.
Sources: NCERT Class 9 Maths: The World of Numbers, locating square roots.
Which rules help you simplify surds?
For positive numbers, √a × √b = √(ab), and √a ÷ √b = √(a/b).
Also, √a × √a = a. So √5 × √5 = 5.
- Simplify by pulling out the largest perfect square: √72 = √36 × √2 = 6√2.
- Like surds have the same surd part, so 3√2 + 5√2 = 8√2.
- Unlike surds, such as √2 and √3, cannot be added into one term.
Sources: NCERT Class 9 Maths: The World of Numbers, irrational numbers; Practice question: root of 5 times root of 5.
What does it mean to rationalise a denominator?
It means removing the surd from the bottom of a fraction. Multiply top and bottom by a suitable surd.
For 6/√3, multiply by √3 on both sides. The bottom becomes 3, and the answer is 2√3.
Sources: NCERT Class 9 Maths: The World of Numbers, working with square roots.
What is a conjugate?
The conjugate of (a + √b) is (a − √b). Their product has no surd: (a + √b)(a − √b) = a² − b.
Use it to rationalise a denominator such as (√5 − √3).
Sources: NCERT Class 9 Maths: The World of Numbers, irrational numbers.
Check 1
The expression √9 is not a surd.
Check 2
√2 + √3 equals √5.
Check 3
(2^3)^2 equals 2^6.
Check 4
Any number raised to the power 0 is 0.
How do you solve five typical problems step by step?
Example 1: Simplify 3^5 × 3^4 ÷ 3^6.
Step 1. The base is the same. Add the indices for the product: 5 + 4 = 9.
Step 2. Subtract the index of the divisor: 9 − 6 = 3.
Step 3. The result is 3^3 = 27.
Check. 243 × 81 = 19683, and 19683 ÷ 729 = 27.
Why this works. Each index counts how many copies of 3 are multiplied. Multiplying adds copies. Dividing removes copies.
Example 2: Simplify (2^3)^4 ÷ 2^10.
Step 1. Power of a power: 3 × 4 = 12. So the top is 2^12.
Step 2. Divide by 2^10. Subtract: 12 − 10 = 2.
Step 3. The result is 2^2 = 4.
Check. 2^12 = 4096 and 2^10 = 1024. Then 4096 ÷ 1024 = 4.
Why this works. Bracketed powers are repeated blocks, so their indices multiply.
Example 3: Find x if 5^(x+1) = 625.
Step 1. Write 625 as a power of 5: 5 × 5 × 5 × 5, so 625 = 5^4.
Step 2. The bases now match, so the indices must match: x + 1 = 4.
Step 3. So x = 3.
Check. 5^(3+1) = 5^4 = 625.
Why this works. Two powers of the same base are equal only when their indices are equal.
Example 4: Simplify √72, and find √12 × √3.
Step 1. The largest perfect square inside 72 is 36. So √72 = √36 × √2 = 6√2.
Step 2. Multiply under one root: √12 × √3 = √36 = 6.
Check. (6√2)^2 = 36 × 2 = 72, so 6√2 is the square root of 72.
Why this works. The product rule lets us split or join numbers under a root. A perfect square then comes out whole.
Example 5: Rationalise 1/(√5 − √3).
Step 1. The conjugate of the bottom is (√5 + √3). Multiply top and bottom by it.
Step 2. The bottom becomes (√5)^2 − (√3)^2 = 5 − 3 = 2.
Step 3. The answer is (√5 + √3)/2.
Check. √5 ≈ 2.236 and √3 ≈ 1.732. The left side is 1/0.504 ≈ 1.984. The right side is 3.968/2 = 1.984.
Why this works. The conjugate product has no surd. So the denominator becomes a plain number.
How do you avoid careless mistakes?
What habits keep answers right?
Look at the bases first. The index laws work only when the bases are equal.
Test a small case. If 2^3 × 2^4 gives 128, then your rule for adding indices is right.
Check 5
2^3 × 3^3 equals 6^6.
Check 6
The surd 3√2 + 5√2 equals 8√2.
Exam traps
Base trap. The laws of indices need the same base. 2^3 × 3^2 does not combine into one power.
Add-or-multiply trap. Product of powers adds indices. Power of a power multiplies them.
Zero trap. Any non-zero number to the power 0 is 1, not 0.
Surd trap. √a + √b is not √(a + b). Unlike surds do not merge.
Rationalise trap. Multiply top and bottom by the same thing, or the value changes.
MCQ on Surds and Indices
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