Exponents and Powers Exercise 13.1 Class 7 Maths NCERT Solutions

What are Exponents and Powers?

Imagine you have a number, say 5, and you need to multiply it by itself four times: 5 x 5 x 5 x 5. Writing this repeatedly can become very long and tedious, especially for larger numbers of multiplications. This is where exponents offer a brilliant solution! An exponent is a shorthand way to represent repeated multiplication of the same number.

In an expression like 5^4:

• The number being multiplied (5) is called the base.
• The small number written at the top right (4) is called the exponent or power.
• The entire expression, 5^4, is read as "5 raised to the power of 4" or simply "5 to the power 4". If the exponent is 2, we say "squared" (e.g., 5^2 is "5 squared"), and if it's 3, we say "cubed" (e.g., 5^3 is "5 cubed").

So, 5^4 literally means 5 x 5 x 5 x 5. The exponent tells you how many times to multiply the base by itself. It's a powerful tool for simplifying expressions and dealing with very large or very small numbers in science and engineering. Understanding this fundamental concept is crucial for mastering Exercise 13.1 and future math topics!

Writing Numbers as a Product of Powers of Prime Factors

1. Understand the Goal — Our goal is to express a given number (like 72) as a multiplication of prime numbers (like 2, 3, 5, 7, etc.) where each prime number is raised to a certain power (exponent).
2. Step 1: Start with Prime Factorization — Begin by finding the prime factors of the given number. Divide the number by the smallest prime factor possible (usually 2), and continue dividing the quotient by that prime factor until it's no longer divisible. Then move to the next prime factor (3, then 5, and so on).
3. Step 2: List the Prime Factors — Write down all the prime factors you found in a multiplication string.
4. Step 3: Group Identical Factors — Group the identical prime factors together.
5. Step 4: Convert to Exponential Form — For each group of identical prime factors, count how many times that prime factor appears. This count will be the exponent for that prime base. For example, if you have 2 x 2 x 2, it becomes 2^3.
6. Example: Express 72 as a product of powers of prime factors — 1. Prime Factorization: 72 ÷ 2 = 36 36 ÷ 2 = 18 18 ÷ 2 = 9 9 ÷ 3 = 3 3 ÷ 3 = 1 2. List Factors: 2 x 2 x 2 x 3 x 3 3. Group Factors: (2 x 2 x 2) x (3 x 3) 4. Exponential Form: 2^3 x 3^2 Thus, 72 in exponential form is 2^3 x 3^2.

Finding the Value of Exponential Expressions

• Example 1: Find the value of 7^3 7^3 means 7 multiplied by itself 3 times. 7 x 7 x 7 = 49 x 7 = 343 So, the value of 7^3 is 343.
• Example 2: Find the value of (-5)^2 (-5)^2 means -5 multiplied by itself 2 times. (-5) x (-5) Remember, a negative number multiplied by a negative number gives a positive number. = 25 So, the value of (-5)^2 is 25.
• Example 3: Find the value of (-2)^3 (-2)^3 means -2 multiplied by itself 3 times. (-2) x (-2) x (-2) = (4) x (-2) = -8 So, the value of (-2)^3 is -8. (Notice that an odd exponent with a negative base results in a negative value.)
• Example 4: Find the value of (1/3)^4 (1/3)^4 means (1/3) multiplied by itself 4 times. (1/3) x (1/3) x (1/3) x (1/3) Multiply the numerators and the denominators separately. = (1 x 1 x 1 x 1) / (3 x 3 x 3 x 3) = 1 / 81 So, the value of (1/3)^4 is 1/81.

Exam Tip: Avoiding Common Mistakes with Exponents

When working with exponents, students often make a few common mistakes. Be careful to avoid these to score well in Exercise 13.1:

1. Confusing Repeated Multiplication with Simple Multiplication: Remember, 2^3 is not 2 x 3. It is 2 x 2 x 2 = 8. Always expand it in your mind if you're unsure.
2. Incorrectly Handling Negative Bases: Pay close attention to parentheses. (-3)^2 means (-3) x (-3) = 9. However, -3^2 means -(3 x 3) = -9. The exponent only applies to the base it's directly attached to. If there are no parentheses, it applies only to the number, not the negative sign.
3. Forgetting Powers of 1: Any number raised to the power of 1 is the number itself (e.g., 7^1 = 7).
4. Miscalculating the Product: Even after correctly expanding, be careful with your multiplication, especially with larger numbers. Double-check your arithmetic.

Practice Questions with Solutions

• Q: Express 6 x 6 x 6 x 6 in exponential form. A: Step 1: Identify the base, which is the number being multiplied repeatedly. Base = 6 Step 2: Count how many times the base is multiplied by itself. This is the exponent. Exponent = 4 (since 6 appears 4 times) Final answer: 6^4
• Q: Find the value of (-4)^3. A: Step 1: Identify the base and the exponent. Base = -4, Exponent = 3 Step 2: Expand the expression to show repeated multiplication. (-4)^3 = (-4) x (-4) x (-4) Step 3: Perform the multiplication carefully. (-4) x (-4) = 16 16 x (-4) = -64 Final answer: -64
• Q: Express 324 as a product of powers of its prime factors. A: Step 1: Start prime factorization by dividing by the smallest prime, 2. 324 ÷ 2 = 162 162 ÷ 2 = 81 Step 2: Continue with the next prime, 3, as 81 is not divisible by 2. 81 ÷ 3 = 27 27 ÷ 3 = 9 9 ÷ 3 = 3 3 ÷ 3 = 1 Step 3: List all prime factors and group them. 2 x 2 x 3 x 3 x 3 x 3 = (2 x 2) x (3 x 3 x 3 x 3) Step 4: Write in exponential form. 2^2 x 3^4 Final answer: 2^2 x 3^4
• Q: Which is greater: 2^5 or 5^2? A: Step 1: Calculate the value of 2^5. 2^5 = 2 x 2 x 2 x 2 x 2 = 32 Step 2: Calculate the value of 5^2. 5^2 = 5 x 5 = 25 Step 3: Compare the two values. 32 > 25 Final answer: 2^5 is greater than 5^2.

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