Quick Revision Notes: Class 7 Mathematics

Chapter: Exponents and Powers (घातांक और घात)

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1. Introduction to Exponents (घातांक का परिचय)

When we need to write very large numbers compactly, we use Exponents.

• Example: $10000 = 10 \times 10 \times 10 \times 10 = 10^4$
• Here, 10 is the Base (आधार) and 4 is the Exponent or Power (घातांक).
• Read as: "10 raised to the power of 4" (10 की घात 4).

In general, for any non-zero rational number $a$ and a positive integer $n$: $$a^n = \underbrace{a \times a \times a \times \dots \times a}_{n \text{ times}}$$

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2. Laws of Exponents (घातांक के नियम)

These are the fundamental rules used to simplify expressions involving exponents. Let $a$ and $b$ be non-zero integers (or rational numbers), and let $m$ and $n$ be whole numbers.

I. Multiplying Powers with the Same Base (समान आधार वाली घातों का गुणन)

To multiply two powers with the same base, add their exponents while keeping the base the same. $$a^m \times a^n = a^{m + n}$$

• Example: $2^3 \times 2^4 = 2^{3 + 4} = 2^7$

II. Dividing Powers with the Same Base (समान आधार वाली घातों का विभाजन)

To divide two powers with the same base, subtract the exponent of the denominator from the exponent of the numerator. $$a^m \div a^n = \frac{a^m}{a^n} = a^{m - n} \quad (\text{where } m > n)$$

• Example: $5^6 \div 5^2 = 5^{6 - 2} = 5^4$

III. Taking a Power of a Power (घात की घात लेना)

To find the power of a power, multiply the exponents while keeping the base the same. $$(a^m)^n = a^{m \times n} = a^{mn}$$

• Example: $(3^2)^4 = 3^{2 \times 4} = 3^8$

IV. Multiplying Powers with the Same Exponents (समान घातांक वाली घातों का गुणन)

If the bases are different but the exponents are the same, multiply the bases and keep the common exponent. $$a^m \times b^m = (a \times b)^m = (ab)^m$$

• Example: $2^3 \times 5^3 = (2 \times 5)^3 = 10^3$

V. Dividing Powers with the Same Exponents (समान घातांक वाली घातों का विभाजन)

If the bases are different but the exponents are the same, divide the bases and keep the common exponent. $$a^m \div b^m = \frac{a^m}{b^m} = \left(\frac{a}{b}\right)^m$$

• Example: $4^3 \div 2^3 = \left(\frac{4}{2}\right)^3 = 2^3$

VI. Number with Exponent Zero (शून्य घात वाली संख्या)

Any non-zero number raised to the power of zero is equal to 1. $$a^0 = 1 \quad (\text{where } a \neq 0)$$

• Example: $7^0 = 1$, $(-5)^0 = 1$, $(1000)^0 = 1$

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3. Negative Exponents (ऋणात्मक घातांक)

A negative exponent indicates a reciprocal. For any non-zero integer $a$ and positive integer $n$: $$a^{-n} = \frac{1}{a^n}$$

• Example: $5^{-2} = \frac{1}{5^2} = \frac{1}{25}$
• Similarly, $\frac{1}{a^{-n}} = a^n$

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4. Decimal Number System / Standard Form (मानक रूप / वैज्ञानिक संकेतन)

Any numbers can be expressed as a decimal number between $1.0$ and $10.0$ (including $1.0$) multiplied by a power of $10$. This is called the Standard Form or Scientific Notation.

• Example 1 (Large Number): $85,00,000 = 8.5 \times 10,00,000 = 8.5 \times 10^6$
• Example 2 (Small Number): $0.000045 = \frac{4.5}{1,00,000} = \frac{4.5}{10^5} = 4.5 \times 10^{-5}$

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5. Quick Tips for Problem Solving (त्वरित सुझाव)

1. Prime Factorization: To express a large composite number in exponential form, first find its prime factors (अभाज्य गुणनखंड).
   • Example: $72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2$
2. Comparing Numbers: Always convert numbers into the same base or write them in standard form to compare which one is larger.
3. Sign Rule for Negative Bases:
   • (Negative base) Even exponent = Positive result (e.g., $(-2)^2 = +4$)
   • (Negative base) Odd exponent = Negative result (e.g., $(-2)^3 = -8$)