Solve for y square root of 2y=10

Solution:

Step 1:

Square both sides to eliminate the square root, resulting in $(\sqrt{2y})^2 = 10^2$.

Step 2:

Proceed to simplify the equation.

Step 2.1:

Express the square root as a power: $(2y)^{\frac{1}{2}}$ becomes $((2y)^{\frac{1}{2}})^2 = 10^2$.

Step 2.2:

Focus on simplifying the left-hand side.

Step 2.2.1:

Consider the expression $((2y)^{\frac{1}{2}})^2$.

Step 2.2.1.1:

Apply exponent multiplication: $((2y)^{\frac{1}{2}})^2$.

Step 2.2.1.1.1:

Invoke the power of a power rule: $(2y)^{\frac{1}{2} \cdot 2} = 10^2$.

Step 2.2.1.1.2:

Eliminate the common factor of 2: $(2y)^{\frac{1}{\cancel{2}} \cdot \cancel{2}} = 10^2$.

Step 2.2.1.1.2.1:

Simplify the expression: $(2y)^1 = 10^2$.

Step 2.2.1.2:

Reduce to $2y = 10^2$.

Step 2.3:

Now, simplify the right-hand side.

Step 2.3.1:

Calculate $10^2$: $2y = 100$.

Step 3:

Divide both sides by 2 to isolate $y$.

Step 3.1:

Divide the equation by 2: $\frac{2y}{2} = \frac{100}{2}$.

Step 3.2:

Simplify the left-hand side.

Step 3.2.1:

Cancel out the 2s: $\frac{\cancel{2} y}{\cancel{2}} = \frac{100}{2}$.

Step 3.2.1.1:

Divide $y$ by 1: $y = \frac{100}{2}$.

Step 3.3:

Simplify the right-hand side.

Step 3.3.1:

Perform the division: $y = 50$.

Knowledge Notes:

The problem-solving process involves algebraic manipulation to solve for $y$ in the equation $\sqrt{2y} = 10$. The steps include:

1. Squaring Both Sides: This is done to remove the square root. According to the property $(\sqrt{a})^2 = a$, squaring the square root of a number returns the original number.

2. Simplification of Expressions: The power rule of exponents is used here, where $(a^m)^n = a^{mn}$. This is applied to simplify the expression after squaring both sides.

3. Dividing Both Sides: To isolate $y$, we divide both sides of the equation by the coefficient of $y$. This is based on the principle of performing the same operation on both sides of the equation to maintain equality.

4. Power of a Power Rule: This rule states that when you raise a power to another power, you multiply the exponents. In this case, raising $(2y)^{\frac{1}{2}}$ to the power of 2 results in $(2y)^{\frac{1}{2} \cdot 2}$, which simplifies to $2y$.

5. Squaring a Number: Squaring a number is the same as raising it to the power of 2. For example, $10^2 = 100$.

6. Division: The final step involves dividing both sides of the equation by the same number to solve for the variable. In this case, dividing by 2 gives the solution for $y$.