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 $y$.