Class 8 Maths Chapter 10 Exponents and Powers Worksheet - Free Printable

Actually, you did not upload an image — the text you provided is a copy-pasted excerpt from a math study guide or textbook page titled “Exponents and Powers”, likely from “Edufever” (as seen in the watermark).

Since there’s no actual *new problem* or *task* embedded in the image (because no image was uploaded), I’ll assume you want me to:

✔ Explain the key concepts in this excerpt
✔ Solve sample problems based on the rules given
✔ Clarify any confusing parts

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## 🔍 Let’s Break Down the Content

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1. Exponential Form of Large Numbers

> Example: `1000 = 10³`

- Base: 10 → the number being multiplied.
- Exponent/Power: 3 → how many times the base is multiplied by itself.
- So, `10³ = 10 × 10 × 10 = 1000`.

✔ This is useful for writing large numbers compactly.

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2. & 3. Behavior of Exponents with Base 10

> As exponent increases by 1 → value becomes 10 times previous.
> As exponent decreases by 1 → value becomes 1/10 of previous.

Examples:
- `10² = 100`
- `10³ = 1000` → 10× bigger than 100
- `10¹ = 10` → 1/10 of 100
- `10⁰ = 1` → 1/10 of 10
- `10⁻¹ = 0.1` → 1/10 of 1

This is the foundation of the decimal system and scientific notation.

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4. Negative Exponents

> For any non-zero integer ‘a’, `a⁻ᵐ = 1/aᵐ`, where m is a natural number.

Example:
- `2⁻³ = 1/2³ = 1/8`
- `10⁻² = 1/100 = 0.01`

✔ This extends exponents to negative integers.

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5. Power of a Fraction

> `(a/b)⁻ᵐ = (b/a)ᵐ`, where m is a natural number.

Example:
- `(2/3)⁻² = (3/2)² = 9/4`

This is consistent with rule #4: `(a/b)⁻ᵐ = 1/(a/b)ᵐ = (b/a)ᵐ`

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6. Laws of Exponents (for rational numbers a, b ≠ 0)

Let’s go one by one:

#### i. `aˣ × aʸ = aˣ⁺ʸ`
> Multiply same base → add exponents.

Example: `2³ × 2⁴ = 2⁷ = 128`

#### ii. `aˣ ÷ aʸ = aˣ⁻ʸ`
> Divide same base → subtract exponents.

Example: `5⁶ ÷ 5² = 5⁴ = 625`

#### iii. `(aˣ)ʸ = aˣʸ`
> Power of a power → multiply exponents.

Example: `(3²)³ = 3⁶ = 729`

#### iv. `(ab)ˣ = aˣ × bˣ`
> Power of a product → distribute exponent.

Example: `(2×3)² = 2² × 3² = 4 × 9 = 36`

#### v. `(a/b)ˣ = aˣ / bˣ`
> Power of a quotient → apply exponent to numerator and denominator.

Example: `(4/2)³ = 4³ / 2³ = 64 / 8 = 8`

#### vi. `(a/b)⁻ᵐ = (b/a)ᵐ`
> Already covered in #5 — same thing.

#### vii. `a⁰ = 1`
> Any non-zero number to the power 0 is 1.

Example: `7⁰ = 1`, `(-5)⁰ = 1`, `(2/3)⁰ = 1`

#### viii. `(-1)ᵒᵈᵈ = -1`, `(-1)ᵉᵛᵉⁿ = 1`
> Useful for simplifying expressions with negative bases.

Examples:
- `(-1)⁵ = -1`
- `(-1)⁴ = 1`

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7. Exponential Equation

> An equation where the unknown is in the exponent.

Example: `5ˣ = 625`

→ Since `625 = 5⁴`, so `x = 4`

Another example from the text: `3^(x+1) = 15` — this is trickier and requires logarithms (not covered here).

But if it’s simple like `2^x = 8`, then since `8 = 2³`, so `x = 3`.

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8. Standard Form / Scientific Notation

> A number is in standard form if written as `k × 10ⁿ`, where:
> - `1 ≤ k < 10`
> - `n` is a positive or negative integer

Examples from text:

#### i. `160000 = 1.6 × 10⁵`
- Move decimal 5 places left → exponent +5
- `k = 1.6` (between 1 and 10)

#### ii. `1548000 = 1.548 × 10⁶`
- Move decimal 6 places left → exponent +6
- `k = 1.548`

✔ This is essential in science and engineering for handling very large or very small numbers.

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## 🧠 Let’s Solve Some Practice Problems Based on This

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✔ Problem 1: Simplify using laws of exponents

> `(2³ × 2⁵) ÷ 2⁴`

Solution:
- First, `2³ × 2⁵ = 2^(3+5) = 2⁸`
- Then, `2⁸ ÷ 2⁴ = 2^(8−4) = 2⁴ = 16`

✔ Answer: 16

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✔ Problem 2: Evaluate

> `(-3)⁴ × (-3)⁻²`

Solution:
- Add exponents: `(-3)^(4 + (-2)) = (-3)² = 9`

✔ Answer: 9

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✔ Problem 3: Write in scientific notation

> `0.00045`

Solution:
- Move decimal 4 places right → exponent = -4
- `k = 4.5`
- So, `4.5 × 10⁻⁴`

✔ Answer: 4.5 × 10⁻⁴

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✔ Problem 4: Solve exponential equation

> `2^(x+3) = 32`

Solution:
- `32 = 2⁵`
- So, `2^(x+3) = 2⁵`
- Therefore, `x + 3 = 5` → `x = 2`

✔ Answer: x = 2

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✔ Problem 5: Simplify

> `(3/5)⁻³`

Solution:
- Use rule: `(a/b)⁻ᵐ = (b/a)ᵐ`
- So, `(3/5)⁻³ = (5/3)³ = 125/27`

✔ Answer: 125/27

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## 💡 Summary

You’ve got a solid overview of:

- Writing large/small numbers with exponents
- Rules for multiplying, dividing, and raising powers
- Negative exponents and fractional bases
- Solving exponential equations
- Scientific notation

These are foundational for algebra, calculus, physics, chemistry, and computer science!

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## ❓ If You Meant Something Else…

If you were expecting me to solve a specific problem from an image you thought you uploaded, please:

➡️ Re-upload the image
➡️ Or paste the exact problem text here

I’m ready to help! 😊

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✔ Final Tip: Memorize the 8 laws — they’re your toolkit for simplifying almost any exponent problem!