Exponents and Powers Ex 13.2 Class 7 NCERT Solutions

Understanding the Laws of Exponents

To master exponents and powers ex 13 2 class 7 ncert, we first need to understand the fundamental rules governing exponents. These rules apply when the bases of the exponential terms are the same, or when the powers are the same. Let's list the five key laws of exponents:

1. Product Law: a^m a^n = a^(m+n). When multiplying terms with the same base, we keep the base and add the exponents. For example, 3^2 3^4 = 3^(2+4) = 3^6.
2. Quotient Law: a^m / a^n = a^(m-n) (where m > n). When dividing terms with the same base, we subtract the exponent of the denominator from the exponent of the numerator.
3. Power of a Power Law: (a^m)^n = a^(m*n). When raising an exponent to another power, we multiply the powers.
4. Product with Same Exponents: a^m * b^m = (ab)^m. If bases are different but powers are the same, we multiply the bases and keep the exponent.
5. Quotient with Same Exponents: a^m / b^m = (a/b)^m. If bases are different but powers are the same during division, we divide the bases and keep the exponent.

Key Terms to Remember

Base

The non-zero integer or rational number that is multiplied by itself repeatedly in an exponential expression. In 5^3, 5 is the base.

Exponent

The number that shows how many times the base is multiplied by itself. Also called the power or index. In 5^3, 3 is the exponent.

Zero Exponent Law

Any non-zero base raised to the power of zero is always equal to 1. That is, a^0 = 1 for any non-zero number a.

Worked Examples of Exercise 13.2 Concepts

• Example 1: Simplify and express in exponential form: (3^2)^4 3^3. Step 1: Apply the 'Power of a Power' law to (3^2)^4. This becomes 3^(2 4) = 3^8. Step 2: Now, multiply 3^8 * 3^3 using the Product Law: 3^(8+3) = 3^11. Final answer: 3^11
• Example 2: Simplify: (2^8 a^5) / (4^3 a^3). Step 1: Express the base 4 as a power of 2: 4^3 = (2^2)^3 = 2^(2 3) = 2^6. Step 2: Rewrite the expression: (2^8 a^5) / (2^6 a^3). Step 3: Apply the Quotient Law to both bases (2 and a): 2^(8-6) a^(5-3) = 2^2 * a^2. Step 4: Use the Product with Same Exponents law: (2a)^2. Final answer: (2a)^2 or 4a^2

Exam Tip: Avoiding Common Exponent Mistakes

Watch out for bases that are different! A common mistake students make in class 7 maths exponents and powers ex 13 2 is trying to add exponents when the bases are not the same. For example, 2^3 3^2 does NOT equal 6^5. You can only apply the product rule a^m a^n = a^(m+n) when the bases (a) are absolutely identical. If they are different, evaluate each term individually or check if you can express them with a common base (e.g., changing 9 to 3^2). Another trap is adding instead of multiplying in (a^m)^n; remember, (5^2)^3 is 5^6, not 5^5!

Practice Questions with Solutions

• Q: Simplify and write the answer in exponential form: 2^3 2^4 2^5 A: Step 1: Identify that the bases are all the same, which is 2. Step 2: Apply the product law of exponents: a^m a^n a^p = a^(m+n+p). Step 3: Add the powers: 3 + 4 + 5 = 12. Final answer: 2^12
• Q: Simplify the expression: (5^2)^3 / 5^3 A: Step 1: Use the Power of a Power rule to simplify (5^2)^3, which gives 5^(2 * 3) = 5^6. Step 2: Now, divide 5^6 / 5^3. Step 3: Apply the quotient law of exponents: a^m / a^n = a^(m-n) for same bases. Step 4: Subtract the exponents: 6 - 3 = 3. Final answer: 5^3
• Q: Simplify: (3 7^2 11^8) / (21 11^3) A: Step 1: Prime factorize 21 in the denominator: 21 = 3 7. Step 2: Rewrite the fraction: (3^1 7^2 11^8) / (3^1 7^1 11^3). Step 3: Simplify the common bases using division rules. Step 4: For base 3: 3^(1-1) = 3^0 = 1. Step 5: For base 7: 7^(2-1) = 7^1 = 7. Step 6: For base 11: 11^(8-3) = 11^5. Step 7: Multiply the simplified terms: 1 7 11^5. Final answer: 7 * 11^5
• Q: Express the following product as a power of prime factors: 108 192 A: Step 1: Find the prime factorization of 108. 108 = 2 54 = 2^2 27 = 2^2 3^3. Step 2: Find the prime factorization of 192. 192 = 2 96 = 2^2 48 = 2^3 24 = 2^4 12 = 2^5 6 = 2^6 3^1. Step 3: Multiply the prime factors: (2^2 3^3) (2^6 3^1). Step 4: Combine powers of same bases: 2^(2+6) 3^(3+1) = 2^8 3^4. Final answer: 2^8 3^4

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