CBSE Class 9 Maths Chapter 1: Number Systems Ex 1.6 - Laws of Exponents

Understanding Exponents and Bases

Before diving into the laws, let's quickly recap what exponents represent. An exponent tells us how many times a base number is multiplied by itself. For example, in 2^3, '2' is the base and '3' is the exponent, meaning 2 × 2 × 2 = 8. In Class 9, we expand this concept to include real numbers as bases and rational numbers as exponents. This means we can have expressions like 64^(1/2) or 32^(2/5). These fractional exponents relate directly to roots; for instance, x^(1/n) is the nth root of x (written as n√x). So, 64^(1/2) is simply the square root of 64, which is 8. Understanding this connection is key to simplifying expressions and solving problems effectively. The laws we're about to explore provide a systematic way to handle these operations, making complex calculations manageable and logical.

Laws of Exponents for Real Numbers

Product Rule

a^m ⋅ a^n = a^(m+n). When multiplying powers with the same base, add the exponents. Example: 2^3 ⋅ 2^4 = 2^(3+4) = 2^7.

Power Rule

(a^m)^n = a^(mn). When raising a power to another power, multiply the exponents. Example: (3^2)^5 = 3^(2⋅5) = 3^10.

Quotient Rule

a^m / a^n = a^(m-n) (where a ≠ 0). When dividing powers with the same base, subtract the exponents. Example: 5^6 / 5^2 = 5^(6-2) = 5^4.

Power of a Product Rule

(ab)^m = a^m ⋅ b^m. When raising a product to a power, apply the exponent to each factor. Example: (2⋅x)^3 = 2^3 ⋅ x^3 = 8x^3.

Zero Exponent Rule

a^0 = 1 (where a ≠ 0). Any non-zero base raised to the power of zero is 1. Example: 7^0 = 1.

Negative Exponent Rule

a^(-n) = 1/a^n (where a ≠ 0). A negative exponent means taking the reciprocal of the base raised to the positive exponent. Example: 4^(-2) = 1/4^2 = 1/16.

Rational Exponent Rule

a^(m/n) = (n√a)^m = n√(a^m). A fractional exponent m/n means taking the nth root of 'a' and then raising it to the power of 'm'. Example: 8^(2/3) = (3√8)^2 = 2^2 = 4.

Applying the Laws of Exponents: Worked Examples

• Example 1: Simplify 64^(1/2) Step 1: Identify the base and exponent. Here, the base is 64 and the exponent is 1/2. Step 2: Relate fractional exponent to roots. We know that a^(1/n) = n√a. So, 64^(1/2) is the square root of 64. Step 3: Calculate the root. √64 = 8. Final Answer: 8
• Example 2: Simplify 32^(2/5) Step 1: Identify the base and exponent. Base = 32, Exponent = 2/5. Step 2: Apply the rational exponent rule: a^(m/n) = (n√a)^m. So, 32^(2/5) = (5√32)^2. Step 3: Calculate the nth root. Find the fifth root of 32. We know that 2^5 = 32, so 5√32 = 2. Step 4: Raise the result to the power m. Now, (5√32)^2 = 2^2 = 4. Final Answer: 4
• Example 3: Simplify 2^(2/3) ⋅ 2^(1/3) Step 1: Identify the bases and exponents. Bases are both 2. Exponents are 2/3 and 1/3. Step 2: Apply the Product Rule: a^m ⋅ a^n = a^(m+n). Since the bases are the same, add the exponents: 2^(2/3 + 1/3). Step 3: Add the fractions. 2/3 + 1/3 = 3/3 = 1. Step 4: Write the simplified expression. 2^1 = 2. Final Answer: 2
• Example 4: Simplify (1/3^5)^4 Step 1: Rewrite the base using a negative exponent (optional but helpful). 1/3^5 can be written as 3^(-5). Step 2: Apply the Power Rule: (a^m)^n = a^(mn). So, (3^(-5))^4 = 3^(-5 * 4) = 3^(-20). Step 3: Convert back to positive exponent if required (standard practice). 3^(-20) = 1/3^20. Final Answer: 1/3^20

Exam Tip: Avoiding Common Mistakes in Exponents

Students often make crucial errors when applying exponent laws. One common mistake is confusing the product rule with the power of a product rule. Remember, a^m ⋅ a^n = a^(m+n) applies when bases are the same, while (ab)^m = a^m ⋅ b^m applies when different bases are multiplied and then raised to a power. Another frequent error is incorrectly handling fractional exponents, especially remembering that a^(1/n) means the nth root, not multiplication by 1/n. Always calculate the root first, or simplify the exponent before performing other operations. Also, be careful with negative signs, especially when the base is negative, e.g., (-2)^3 vs. -2^3. Practice writing down each step clearly to avoid careless errors in calculations, particularly with fractions.

Practice Questions with Solutions

• Q: Simplify: 125^(1/3) A: Step 1: Identify the base (125) and the exponent (1/3). Step 2: Apply the rational exponent rule: a^(1/n) = n√a. So, 125^(1/3) = 3√125. Step 3: Find the cube root of 125. We know that 5^3 = 125, so 3√125 = 5. Final answer: 5
• Q: Find the value of 9^(3/2) A: Step 1: Identify the base (9) and the exponent (3/2). Step 2: Apply the rational exponent rule: a^(m/n) = (n√a)^m. So, 9^(3/2) = (√9)^3. Step 3: Calculate the square root of 9: √9 = 3. Step 4: Raise the result to the power of 3: 3^3 = 3 × 3 × 3 = 27. Final answer: 27
• Q: Simplify: 7^(1/2) ⋅ 8^(1/2) A: Step 1: Identify the bases (7 and 8) and the exponents (1/2 and 1/2). Step 2: Apply the Power of a Product Rule: a^m ⋅ b^m = (ab)^m. Here, the exponents are the same. Step 3: Multiply the bases and keep the exponent: (7 ⋅ 8)^(1/2) = 56^(1/2). Step 4: Relate the fractional exponent to a root: 56^(1/2) = √56. Final answer: √56 (or 2√14 by simplifying the surd)
• Q: Simplify: 5^(2/3) / 5^(1/3) A: Step 1: Identify the base (5) and the exponents (2/3 and 1/3). Step 2: Apply the Quotient Rule: a^m / a^n = a^(m-n). Since the bases are the same, subtract the exponents. Step 3: Subtract the fractions: 2/3 - 1/3 = 1/3. Step 4: Write the simplified expression: 5^(1/3). Final answer: 5^(1/3) (or 3√5)
• Q: Simplify: ( (1/2)^(-2) ⋅ (1/3)^(-2) ) / (1/6)^(-2) A: Step 1: Apply the negative exponent rule a^(-n) = 1/a^n or (1/a)^(-n) = a^n. (1/2)^(-2) = 2^2 = 4. (1/3)^(-2) = 3^2 = 9. (1/6)^(-2) = 6^2 = 36. Step 2: Substitute these values back into the expression: (4 ⋅ 9) / 36. Step 3: Perform the multiplication: 36 / 36. Step 4: Perform the division. Final answer: 1

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