Rational Exponents (Positive Real Bases) MCQs Quiz | Class 9

This quiz covers Class IX Mathematics (Code 041), Unit I: Number Systems, specifically focusing on Rational Exponents with Positive Real Bases. It tests your understanding of the meaning of a^(p/q) and the application of laws of exponents for rational powers through various cases. Complete the quiz to view your score and download the PDF answer sheet.

Understanding Rational Exponents

In the number system, exponents are not limited to integers. We can have rational numbers (fractions) as exponents. For any positive real number a and rational number p/q (where p and q are integers and q > 0), the expression a^(p/q) represents the q-th root of a raised to the power p.

Key Definition

The form a^(1/n) means the n-th root of a. Therefore:

  • a^(p/q) = (a^p)^(1/q) which is the q-th root of a^p.
  • Alternatively, a^(p/q) = (a^(1/q))^p which is the (q-th root of a) raised to the power p.

Laws of Exponents for Real Numbers

If a, b are positive real numbers and m, n are rational numbers, the following laws hold true:

Law Name Formula Example
Product Law a^m * a^n = a^(m+n) 2^(1/3) * 2^(2/3) = 2^1 = 2
Quotient Law a^m / a^n = a^(m-n) 5^(3/2) / 5^(1/2) = 5^1 = 5
Power of a Power (a^m)^n = a^(mn) (3^2)^(1/2) = 3^1 = 3
Power of a Product (ab)^m = a^m * b^m (2*3)^2 = 4*9 = 36

Quick Revision Notes

  • Negative Exponents: a^(-n) = 1 / a^n. For example, 4^(-1/2) = 1 / 4^(1/2) = 1/2.
  • Zero Exponent: a^0 = 1 (for a not equal to 0).
  • Always try to express the base as a power of a prime number to simplify calculations (e.g., write 8 as 2^3, 27 as 3^3).

Extra Practice Questions

  1. Evaluate: 625^(-1/4).
  2. Simplify: (32/243)^(-4/5).
  3. Find the value of x if 5^(x-3) * 3^(2x-8) = 225.
  4. Simplify: [ (81)^(-1/2) ]^(-1/4) ]^2.
  5. Prove that (x^a / x^b)^(a+b) * (x^b / x^c)^(b+c) * (x^c / x^a)^(c+a) = 1.