Solve for x 10 square root of 2 = square root of 2x

Solution:

Step 1:

Express the equation in its initial form: $\sqrt{2x} = 10\sqrt{2}$.

Step 2:

Eliminate the square root by squaring both sides: $(\sqrt{2x})^2 = (10\sqrt{2})^2$.

Step 3:

Expand and simplify both sides of the equation.

Step 3.1:

Convert the square root to an exponent: $(2x)^{\frac{1}{2}}$.

Step 3.2:

Simplify the left-hand side.

Step 3.2.1:

Square the exponentiated form: $((2x)^{\frac{1}{2}})^2$.

Step 3.2.1.1:

Apply the exponent multiplication rule: $(2x)^{\frac{1}{2} \cdot 2}$.

Step 3.2.1.1.1:

Utilize the power rule $(a^m)^n = a^{mn}$: $(2x)^{\frac{1}{2} \cdot 2}$.

Step 3.2.1.1.2:

Simplify by removing the common factor: $(2x)^{1}$.

Step 3.2.1.2:

Finalize the left side: $2x = (10\sqrt{2})^2$.

Step 3.3:

Simplify the right-hand side.

Step 3.3.1:

Square the term: $(10\sqrt{2})^2$.

Step 3.3.1.1:

Apply the product rule to square $10\sqrt{2}$: $2x = 10^2(\sqrt{2})^2$.

Step 3.3.1.2:

Convert the square root to an exponent: $2x = 100(2^{\frac{1}{2}})^2$.

Step 3.3.1.2.1:

Apply the power rule: $2x = 100 \cdot 2^{\frac{1}{2} \cdot 2}$.

Step 3.3.1.2.2:

Simplify the exponent: $2x = 100 \cdot 2^{1}$.

Step 3.3.1.2.3:

Calculate the product: $2x = 200$.

Step 4:

Isolate $x$ by dividing both sides by $2$: $\frac{2x}{2} = \frac{200}{2}$.

Step 4.1:

Simplify both sides: $x = \frac{200}{2}$.

Step 4.2:

Compute the division: $x = 100$.

Knowledge Notes:

To solve the equation $\sqrt{2x} = 10\sqrt{2}$, we follow these steps:

1. Square Root Properties: The square root of a number $a$, denoted $\sqrt{a}$, can also be written as $a^{\frac{1}{2}}$. Squaring the square root of a number will return the original number, i.e., $(\sqrt{a})^2 = a$.

2. Squaring Both Sides: When an equation contains a square root, one way to solve for the variable is to square both sides of the equation. This eliminates the square root and makes it easier to solve for the variable.

3. Exponent Rules: The power rule for exponents states that $(a^m)^n = a^{mn}$. When an exponent is raised to another exponent, you multiply the exponents.

4. Simplifying Expressions: After applying exponent rules, it's important to simplify the expression by performing any possible multiplications or divisions.

5. Isolating the Variable: To solve for the variable, we need to isolate it on one side of the equation. This often involves dividing both sides of the equation by a number or performing other algebraic operations to get the variable alone.

6. Product Rule: When squaring a product, such as $(ab)^2$, you can square each factor separately, resulting in $a^2b^2$.

By applying these principles, we can solve the given square root equation step by step.