Class 7 Maths Exponents and Powers Notes

Introduction:

The chapter “Exponents and Powers” in Class 7 Maths focuses on the concept of exponents, a fundamental mathematical operation. Exponents are used to express repeated multiplication of the same number. This chapter explains the rules for working with exponents, how to calculate powers, and the application of exponents in various mathematical problems. Understanding exponents is essential as they are used extensively in algebra, science, and real-life situations.

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Key Concepts Covered:

1. What is an Exponent?

• An exponent is a small number written to the top right of a base number that shows how many times the base number is multiplied by itself.
• Example:
  In $2^3$23, 2 is the base, and 3 is the exponent or power. This means $2 \times 2 \times 2 = 8$2×2×2=8.

2. What is a Power?

• The power is the result of raising a base number to an exponent.
• Example:
  $3^4 = 3 \times 3 \times 3 \times 3 = 81$34=3×3×3×3=81.

3. Understanding the Base and Exponent:

• Base: The number that is repeatedly multiplied.
• Exponent: The number of times the base is multiplied by itself.

Example:
In $5^3$53, 5 is the base and 3 is the exponent. This means:$$5^3 = 5 \times 5 \times 5 = 125$$53=5×5×5=125

4. Laws of Exponents:

The laws of exponents are rules that help simplify expressions with exponents. Here are the basic laws:

Product Law:

When multiplying two powers with the same base, add their exponents.$$a^m \times a^n = a^{m+n}$$am×an=am+n

Example:$$2^3 \times 2^2 = 2^{3+2} = 2^5 = 32$$23×22=23+2=25=32

Quotient Law:

When dividing two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator.$$\frac{a^m}{a^n} = a^{m-n}$$anam​=am−n

Example:$$\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625$$5256​=56−2=54=625

Power of a Power:

When raising a power to another power, multiply the exponents.$$(a^m)^n = a^{m \times n}$$(am)n=am×n

Example:$$(3^2)^3 = 3^{2 \times 3} = 3^6 = 729$$(32)3=32×3=36=729

Power of a Product:

When raising a product to a power, apply the exponent to each factor.$$(ab)^n = a^n \times b^n$$(ab)n=an×bn

Example:$$(2 \times 3)^2 = 2^2 \times 3^2 = 4 \times 9 = 36$$(2×3)2=22×32=4×9=36

Power of a Quotient:

When raising a quotient to a power, apply the exponent to both the numerator and the denominator.$$\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}$$(ba​)n=bnan​

Example:$$\left( \frac{2}{3} \right)^2 = \frac{2^2}{3^2} = \frac{4}{9}$$(32​)2=3222​=94​

5. Negative Exponents:

• A negative exponent represents the reciprocal of the base raised to the opposite positive exponent.

$$a^{-m} = \frac{1}{a^m}$$a−m=am1​

Example:$$3^{-2} = \frac{1}{3^2} = \frac{1}{9}$$3−2=321​=91​

6. Zero Exponent:

• Any non-zero number raised to the power of zero is equal to 1.

$$a^0 = 1 \quad \text{(where \( a \neq 0 \))}$$a0=1(where a=0)

Example:$$5^0 = 1$$50=1

7. Standard Form (Scientific Notation):

• Standard form is a way of writing large or small numbers using powers of 10.
• A number is written as the product of a number between 1 and 10, and a power of 10.

$$\text{Example}: 3400 = 3.4 \times 10^3$$Example:3400=3.4×103

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Important Questions with Answers:

Answer:
$0.0005 = 5 \times 10^{-4}$0.0005=5×10−4

What is 242^424?

Answer:
$2^4 = 2 \times 2 \times 2 \times 2 = 16$24=2×2×2×2=16

Simplify 32×333^2 \times 3^332×33.

Answer:
Using the product law:
$3^2 \times 3^3 = 3^{2+3} = 3^5 = 243$32×33=32+3=35=243

Simplify 5652\frac{5^6}{5^2}5256​.

Answer:
Using the quotient law:
$\frac{5^6}{5^2} = 5^{6-2} = 5^4 = 625$5256​=56−2=54=625

What is the value of 404^040?

Answer:
Any non-zero number raised to the power of zero is 1:
$4^0 = 1$40=1

Simplify (2×5)3(2 \times 5)^3(2×5)3.

Answer:
Using the power of a product law:
$(2 \times 5)^3 = 2^3 \times 5^3 = 8 \times 125 = 1000$(2×5)3=23×53=8×125=1000

What is the value of 10−310^{-3}10−3?

Answer:
Using the negative exponent rule:
$10^{-3} = \frac{1}{10^3} = \frac{1}{1000} = 0.001$10−3=1031​=10001​=0.001

Express 0.0005 in standard form.