formula for simplifying square roots, cube roots, and also for the chapter linear equations in one variable (accordig to class 8 cbse 2024)

Here are the formulas and basic rules for simplifying square roots, cube roots, and solving linear equations in one variable, aligned with the **Class 8 CBSE curriculum for 2024**:

### 1. **Simplifying Square Roots**:
To simplify square roots, use the following formula:

\[
\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}
\]

**Steps to simplify square roots**:
- Factor the number inside the square root into its prime factors.
- Group the prime factors into pairs.
- Take one factor out of the square root for each pair.

For example:
\[
\sqrt{72} = \sqrt{2^3 \cdot 3^2} = 3 \cdot \sqrt{2^3} = 6\sqrt{2}
\]

### 2. **Simplifying Cube Roots**:
To simplify cube roots, use the following formula:

\[
\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}
\]

**Steps to simplify cube roots**:
- Factor the number inside the cube root into prime factors.
- Group the prime factors into triples.
- Take one factor out of the cube root for each group of three.

For example:
\[
\sqrt[3]{64} = \sqrt[3]{2^6} = 2^2 = 4
\]

### 3. **Linear Equations in One Variable**:
A **linear equation in one variable** has the general form:

\[
ax + b = c
\]

**Steps to solve**:
1. Simplify both sides of the equation (if necessary).
2. Move all terms containing the variable to one side of the equation.
3. Move constant terms to the opposite side of the equation.
4. Solve for the variable by dividing or multiplying.

For example:
\[
3x + 5 = 14
\]
- Subtract 5 from both sides:
\[
3x = 9
\]
- Divide both sides by 3:
\[
x = 3
\]

### Summary of Important Formulas:
1. **Square Root**: \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)
2. **Cube Root**: \(\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}\)
3. **Linear Equation**: \(ax + b = c \rightarrow x = \frac{c - b}{a}\)

---

Do you want further details on any of these topics or have any questions?

Here are 5 related questions you might find helpful:
1. How do we solve equations involving square roots on both sides?
2. What are some examples of cube roots that cannot be simplified exactly?
3. How can we handle equations with fractions in linear equations?
4. What happens if a linear equation has no solution or infinite solutions?
5. How do we simplify nested square roots, like \(\sqrt{\sqrt{50}}\)?

**Tip**: When solving linear equations, always check the solution by substituting the value back into the original equation.

formula for simplifying square roots, cube roots, and also for the chapter linear equations in one variable (according to class 8 cbse 2024)

Learn the step-by-step methods for simplifying square roots, cube roots, and solving linear equations in one variable, aligned with the Class 8 CBSE 2024 curriculum. This guide covers important formulas and prime factorization rules, suitable for Class 8 students.

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