RD Sharma Class 9 Maths Exponents of Real Numbers Solutions

According to the CBSE syllabus for the 2025-26 session, the key laws of exponents for real numbers that you must master are:

• Product of Powers Rule: a^m × aⁿ = a^m+n
• Quotient of Powers Rule: a^m / aⁿ = a^m-n
• Power of a Power Rule: (a^m)ⁿ = a^mn
• Power of a Product Rule: (ab)^m = a^mb^m
• Power of a Fraction Rule: (a/b)^m = a^m / b^m
• Zero Exponent Rule: a⁰ = 1 (where a ≠ 0)
• Negative Exponent Rule: a^-m = 1/a^m

Understanding rational exponents is a critical skill for this chapter. A rational exponent, such as a^p/q, is essentially a combination of a root and a power. It can be interpreted as the q-th root of a, raised to the power of p ( (^q√a)^p ). This concept is crucial for simplifying complex expressions involving radicals (surds) and powers together. For example, to solve 81^3/4, you first find the 4th root of 81 (which is 3) and then cube the result (3³ = 27). This two-step approach simplifies otherwise complicated calculations.

A very common mistake is to incorrectly apply the product or quotient rules to terms with different bases. For instance, students might try to simplify an expression like 2³ × 5² by adding the exponents, which is wrong. The rules a^m × aⁿ = a^m+n and a^m / aⁿ = a^m-n are only valid when the bases ('a') are the same. The correct approach for different bases is to calculate each term individually (e.g., 8 × 25 = 200) before performing the multiplication or division.

The exercises in RD Sharma's chapter on Exponents of Real Numbers are designed to provide comprehensive practice. You will typically encounter the following types of problems:

• Direct Simplification: Problems that require the direct application of one or more laws of exponents.
• Evaluating Expressions: Finding the numerical value of expressions with integral and rational exponents.
• Proving Identities: Questions where you need to prove that the Left Hand Side (LHS) equals the Right Hand Side (RHS) using exponent rules.
• Solving for Variables: Finding the value of a variable (like 'x') in exponential equations, for example, 2^x+1 = 16.

Surds and exponents are fundamentally linked. A surd, or a radical, is simply another way of writing a rational exponent. For example, the square root of x (√x) is identical to x^1/2, and the cube root of y (³√y) is the same as y^1/3. The process of 'rationalisation' often uses the laws of exponents to eliminate a surd from the denominator of a fraction, making the expression simpler to work with.

Using both NCERT and RD Sharma provides a balanced and thorough preparation strategy. The NCERT textbook is essential for understanding the core concepts and fundamental principles as per the CBSE curriculum. On the other hand, RD Sharma offers extensive practice with a large volume of questions of varying difficulty. This helps reinforce the concepts, improve problem-solving speed, and gain confidence in tackling any type of question that might appear in the exam.

The zero exponent rule, which states that any non-zero number raised to the power of zero is 1 (a⁰ = 1), is a powerful tool for simplification. In complex algebraic fractions or multi-term expressions, you might encounter a term like (2x²y⁵ / 7z)⁰. Instead of calculating the complex term inside the bracket, you can immediately replace the entire expression with 1, significantly simplifying the problem and reducing the chances of calculation errors.