Class 8 Math Exponents and Powers Notes

sBagul

2 days ago

Exponents and Powers – Class 8

Hi everyone! This chapter is all about Exponents and Powers. They are a shorthand way of writing repeated multiplication and are incredibly useful in many areas of math and science.

What are Exponents and Powers?

When a number is multiplied by itself several times, we can write it in a shorter form using exponents. The number being multiplied is called the base, and the number of times it’s multiplied is called the exponent or power.

Example: 2 × 2 × 2 = 2³ (2 is the base, 3 is the exponent). We say “2 to the power of 3” or “2 cubed”.

Laws of Exponents

Here are some important rules that make working with exponents easier:

1. Product Rule: a^m × aⁿ = a^m+n (When multiplying powers with the same base, add the exponents.)

Application: Simplifying expressions like x² * x³ = x⁵. In computer science, this helps determine the space complexity of algorithms.
2. Quotient Rule: a^m ÷ aⁿ = a^m-n (When dividing powers with the same base, subtract the exponents.)

Application: Simplifying expressions like y⁵ / y² = y³. This is used in physics, such as calculating ratios of forces or intensities.
3. Power Rule: (a^m)ⁿ = a^m×n (When raising a power to another power, multiply the exponents.)

Application: Simplifying expressions like (z²)³ = z⁶. In finance, this is used in compound interest calculations.
4. Product Power Rule: (ab)^m = a^mb^m (A power of a product is the product of the powers.)

Application: Simplifying expressions like (2x)³ = 2³x³ = 8x³. Useful in geometry when scaling dimensions of a shape.
5. Quotient Power Rule: (a/b)^m = a^m/b^m (A power of a quotient is the quotient of the powers.)

Application: Simplifying expressions like (x/y)² = x²/y². Used in chemistry when dealing with ratios and concentrations.
6. Zero Exponent: a⁰ = 1 (Any non-zero number raised to the power of 0 is 1.)

Application: Used as a base case in mathematical induction proofs and in simplifying algebraic expressions.
7. Negative Exponent: a^-n = 1/aⁿ (A negative exponent means the reciprocal of the base raised to the positive exponent.)

Application: Simplifying expressions like x⁻² = 1/x². Used in physics for expressing very small quantities, like wavelengths or distances.

Applications of Exponents and Powers

1. Science (e.g., Physics, Chemistry):

Expressing very large or very small numbers (scientific notation), calculating exponential growth or decay.

2. Computer Science:

Measuring computer memory (kilobytes, megabytes, gigabytes), calculating algorithm complexity.

3. Finance:

Calculating compound interest, population growth, and other exponential changes.

4. Everyday Life:

Understanding scales on maps, calculating areas and volumes, and many other practical uses.

Exponents and powers are fundamental tools in mathematics and are essential for understanding many scientific and real-world phenomena.

Exponents and Powers Quiz – Tough Application Problems

1. **Bacterial Growth:** A bacteria culture doubles in size every hour. If it starts with 1000 bacteria, how many bacteria will there be after 5 hours?

32,000

After 1 hour: 1000 * 2 = 2000. After 2 hours: 2000 * 2 = 4000. In general, after ‘n’ hours, the number of bacteria will be 1000 * 2ⁿ. After 5 hours: 1000 * 2⁵ = 1000 * 32 = 32,000.

2. **Compound Interest:** Rs. 5000 is invested at a compound interest rate of 8% per annum. What will be the amount after 3 years?

Rs. 6298.56

Amount = Principal * (1 + Rate/100)^Time. Amount = 5000 * (1 + 8/100)³ = 5000 * (1.08)³ = 5000 * 1.259712 = Rs. 6298.56.

3. **Population Growth:** The population of a city increases by 5% every year. If the current population is 200,000, what will be the population after 2 years?

220,500

Population after ‘n’ years = Initial Population * (1 + Rate/100)ⁿ. Population after 2 years = 200000 * (1 + 5/100)² = 200000 * (1.05)² = 200000 * 1.1025 = 220,500.

4. **Exponential Decay:** A radioactive substance decays at a rate of 10% per hour. If there are initially 500 grams of the substance, how much will remain after 4 hours?

328.05 grams

Amount remaining after ‘n’ hours = Initial Amount * (1 – Rate/100)ⁿ. Amount after 4 hours = 500 * (1 – 10/100)⁴ = 500 * (0.9)⁴ = 500 * 0.6561 = 328.05 grams.

5. **Computer Memory:** A computer’s memory doubles every year. If it starts with 4 GB of memory, how much memory will it have after 3 years?

32 GB

Memory after ‘n’ years = Initial Memory * 2ⁿ. Memory after 3 years = 4 * 2³ = 4 * 8 = 32 GB.

6. **Scaling a Cube:** If the side of a cube is tripled, how many times greater does its volume become?

27 times greater

Original volume = s³. New side = 3s. New volume = (3s)³ = 27s³. The new volume is 27 times the original volume.

7. **Area of a Square:** The side of a square is given by the expression 2x³. What is the area of the square?

4x⁶

Area of a square = side² = (2x³)² = 4x⁶.

8. **Simplifying a Complex Expression:** Simplify: (a⁴b⁻²)³ * (a⁻¹b⁵)²

a¹⁰b⁴

(a⁴b⁻²)³ * (a⁻¹b⁵)² = a¹²b⁻⁶ * a⁻²b¹⁰ = a¹²⁻²b⁻⁶⁺¹⁰ = a¹⁰b⁴.

9. **Scientific Notation:** The distance to a star is approximately 1.5 x 10¹¹ meters. If a spacecraft travels at a speed of 3 x 10⁴ meters/second, how many seconds will it take to reach the star?

5 x 10⁶ seconds

Time = Distance / Speed = (1.5 x 10¹¹) / (3 x 10⁴) = (1.5/3) * (10¹¹/10⁴) = 0.5 * 10⁷ = 5 x 10⁶ seconds.

10. **Nested Exponents:** Simplify: [(x²)³]⁴

x²⁴

[(x²)³]⁴ = (x⁶)⁴ = x²⁴.