Rational exponents: Class 9 practice

What you should know first

• Laws of exponents
• Roots

The idea behind rational exponents

a^(m/n) means the n-th root of a, raised to the power m. Laws of exponents still hold: a^p × a^q = a^(p+q) and (a^p)^q = a^(pq).

Step-by-step method

1. Rewrite the base as a power of a prime if possible.
2. Multiply exponents for powers of powers.
3. Simplify the resulting whole-number power.

Worked example 1 (Basic)

Find (125)^(1/3).

1. 125 = 5^3
2. (5^3)^(1/3) = 5^1 = 5
3. Answer: 5

Worked example 2 (Standard)

Find (27)^(1/3).

1. 27 = 3^3
2. (3^3)^(1/3) = 3^1 = 3
3. Answer: 3

Worked example 3 (Challenge)

Find (4)^(−2/2).

1. 4 = 2^2
2. (2^2)^(−2/2) = 2^−2
3. = 1/4
4. Answer: 1/4

Common mistakes and misconceptions

• Dividing instead of multiplying exponents
• Taking the power before the root wrongly

Practice questions

1. (Basic) Find (4)^(1/2).
2. (Basic) Find (25)^(1/2).
3. (Standard) Find (9)^(3/2).
4. (Standard) Find (4)^(2/2).
5. (Challenge) Find (8)^(−2/3).
6. (Challenge) Find (125)^(−1/3).

Answer key with full working

• Q1: 4 = 2^2 → (2^2)^(1/2) = 2^1 = 2 → Answer: 2
• Q2: 25 = 5^2 → (5^2)^(1/2) = 5^1 = 5 → Answer: 5
• Q3: 9 = 3^2 → (3^2)^(3/2) = 3^3 = 27 → Answer: 27
• Q4: 4 = 2^2 → (2^2)^(2/2) = 2^2 = 4 → Answer: 4
• Q5: 8 = 2^3 → (2^3)^(−2/3) = 2^−2 → = 1/4 → Answer: 1/4
• Q6: 125 = 5^3 → (5^3)^(−1/3) = 5^−1 → = 1/5 → Answer: 1/5

Syllabus note

This skill is listed under "Number Systems" for Class 9 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.

Related practice and notes

• Number Systems chapter notes (Class 9)
• Rationalising the denominator
• Recurring decimals to fractions
• Simplifying surds
• Remainder theorem and factor theorem
• Area of a triangle by Heron's formula
• Factorising by splitting the middle term
• Cube identities
• All Class 9 practice

Frequently asked questions

Which chapter is rational exponents in?

"Number Systems", Class 9 Maths (NCERT/CBSE). Other boards may place it differently.

How were the answers checked?

Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.