Chapter 12: Exponents and Powers 

Introduction to Exponents and Power

Exponents and Power are a fundamental concept in mathematics that simplify the representation of repeated multiplication. They are used to express large numbers compactly and are essential for understanding various mathematical and scientific applications. In this chapter, we explore the concept of exponents, their properties, and their applications, as outlined in the CBSE Class 8 Mathematics syllabus.

An exponent indicates how many times a number, called the base, is multiplied by itself. For example, in (2^3), the base is 2, and the exponent is 3, meaning (2 \times 2 \times 2 = 8). The expression (2^3) is read as “2 raised to the power of 3” or “2 cubed.”

Key Terms

• Base: The number that is multiplied repeatedly.

• Exponent: The number that indicates how many times the base is multiplied.

• Power: The entire expression, including the base and exponent (e.g., (2^3) is a power).

Powers with Positive Exponents

A positive exponent indicates the number of times the base is multiplied by itself. For instance:

• (3^2 = 3 \times 3 = 9)

• (5^4 = 5 \times 5 \times 5 \times 5 = 625)

This is straightforward for positive integers. The general form is (a^n), where (a) is the base and (n) is the exponent, resulting in (a \times a \times \ldots \times a) (n times).

Powers with Negative Exponents

Negative exponents represent the reciprocal of the base raised to the corresponding positive exponent. The rule is: [ a^{-n} = \frac{1}{a^n}, \text{ where } a \neq 0 ] For example:

• (2^{-3} = \frac{1}{2^3} = \frac{1}{8})

• (5^{-2} = \frac{1}{5^2} = \frac{1}{25})

This rule allows us to handle division involving exponents and simplifies calculations with very small numbers, such as in scientific notation.

Laws of Exponents

The laws of exponents govern how to manipulate expressions involving powers. These laws apply to exponents with the same base or when multiplying/dividing powers. The key laws are:

1. Product of Powers: When multiplying two powers with the same base, add the exponents. [ a^m \times a^n = a^{m+n} ] Example: (2^3 \times 2^2 = 2^{3+2} = 2^5 = 32)

2. Quotient of Powers: When dividing two powers with the same base, subtract the exponents. [ \frac{a^m}{a^n} = a^{m-n}, \text{ where } a \neq 0 ] Example: (\frac{5^4}{5^2} = 5^{4-2} = 5^2 = 25)

3. Power of a Power: When raising a power to another power, multiply the exponents. [ (a^m)^n = a^{m \times n} ] Example: ((3^2)^3 = 3^{2 \times 3} = 3^6 = 729)

4. Power of a Product: The exponent applies to each factor in a product. [ (a \times b)^n = a^n \times b^n ] Example: ((2 \times 3)^2 = 2^2 \times 3^2 = 4 \times 9 = 36)

5. Power of a Quotient: The exponent applies to both the numerator and denominator. [ \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}, \text{ where } b \neq 0 ] Example: (\left(\frac{2}{5}\right)^3 = \frac{2^3}{5^3} = \frac{8}{125})

6. Zero Exponent: Any non-zero number raised to the power of zero is 1. [ a^0 = 1, \text{ where } a \neq 0 ] Example: (7^0 = 1)

7. Negative Exponent Rule: As mentioned earlier, (a^{-n} = \frac{1}{a^n}).

These laws simplify complex calculations and are widely used in algebra and other mathematical operations.

Use of Exponents to Express Small Numbers

Exponents are particularly useful for expressing very large or very small numbers in a compact form, such as in scientific notation. Scientific notation represents numbers as a product of a number between 1 and 10 and a power of 10.

For example:

• Large numbers: (360000 = 3.6 \times 10^5)

• Small numbers: (0.00045 = 4.5 \times 10^{-4})

This is especially useful in science and engineering, where measurements often involve very large or very small quantities, such as distances between planets or sizes of microscopic particles.

Steps to Convert to Scientific Notation

1. Move the decimal point to get a number between 1 and 10.

2. Count the number of places the decimal point was moved.

3. If the original number is greater than 1, the exponent is positive (equal to the number of places moved).

4. If the original number is less than 1, the exponent is negative (equal to the number of places moved).

Example:

• (75000): Move the decimal 4 places left to get 7.5. Thus, (75000 = 7.5 \times 10^4).

• (0.0082): Move the decimal 3 places right to get 8.2. Thus, (0.0082 = 8.2 \times 10^{-3}).

Standard Form

The standard form of a number is its scientific notation, where the number is written as (a \times 10^n), with (1 \leq a < 10) and (n) being an integer. This form is widely used to simplify calculations and comparisons of large or small numbers.

Applications of Exponents

Exponents have practical applications in various fields:

• Science: To express quantities like the speed of light ((3 \times 10^8 , \text{m/s})) or the mass of an electron ((9.1 \times 10^{-31} , \text{kg})).

• Finance: In calculating compound interest, where the formula (A = P(1 + \frac{r}{n})^{nt}) uses exponents.

• Computing: In algorithms and data storage, where powers of 2 are common (e.g., (2^{10} = 1024) for a kilobyte).

• Geometry: In formulas for area and volume, such as the area of a square ((s^2)) or volume of a cube ((s^3)).

Comparing Numbers in Exponential Form

To compare numbers in scientific notation:

1. Compare the coefficients (the numbers between 1 and 10).

2. If the coefficients are equal, compare the exponents.

Example:

• Compare (4.5 \times 10^6) and (7.2 \times 10^5).

  • The exponents are 6 and 5. Since (10^6 > 10^5), (4.5 \times 10^6 > 7.2 \times 10^5).

Common Mistakes to Avoid

1. Misinterpreting Negative Exponents: Ensure that (a^{-n} = \frac{1}{a^n}), not a negative number.

2. Incorrect Application of Laws: For example, (a^m \times a^n \neq a^{m \times n}). Use the correct law ((a^{m+n})).

3. Zero Base with Negative Exponent: Remember that (0^{-n}) is undefined since division by zero is not possible.

4. Forgetting the Zero Exponent Rule: Any non-zero number raised to zero is 1, not 0.

Practice Questions

1. Simplify: (2^4 \times 2^3).

2. Express (0.000067) in scientific notation.

3. Find the value of ((3^2)^3).

4. Compare (6.8 \times 10^4) and (2.3 \times 10^5).

5. Simplify: (\frac{5^7}{5^4}).

Summary

Chapter 12 introduces exponents as a powerful tool for simplifying repeated multiplication and expressing large or small numbers. The laws of exponents provide a systematic way to manipulate expressions, while scientific notation facilitates handling extreme values. Understanding these concepts is crucial for further studies in mathematics and their applications in real-world scenarios.

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Chapter 13: Direct and Inverse Proportions

 

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