Calculating Square Roots: A Step-by-Step Solution

Let's break down this problem into two main steps to find the required positive square root. First, we need to calculate the value of $\alpha$ based on the given expression. Second, we will use the calculated $\alpha$ to find the positive square root of $8 + 2\alpha$.

Step 1: Determining the Value of Alpha ($\alpha$)

The problem states that $\alpha$ is the positive square root of the expression $(5 + 3\sqrt{2})(5 - 3\sqrt{2})$.

Simplifying the Product

The expression inside the square root is $(5 + 3\sqrt{2})(5 - 3\sqrt{2})$. This is in the form of $(a+b)(a-b)$.

We know the algebraic identity: $ (a+b)(a-b) = a^2 - b^2 $.

In this case, $a = 5$ and $b = 3\sqrt{2}$.

Applying the identity, the product is:

Now, let's calculate each term:

• $5^2 = 5 \times 5 = 25$
• $ (3\sqrt{2})^2 = 3^2 \times (\sqrt{2})^2 = 9 \times 2 = 18 $

So, the product is:

Finding the Positive Square Root of Alpha

We are told that $\alpha$ is the positive square root of this product (which is 7).

So, we have found the value of $\alpha$ as $\sqrt{7}$.

Step 2: Calculating the Positive Square Root of \( 8 + 2\alpha \)

Now we need to find the positive square root of $8 + 2\alpha$. We substitute the value of $\alpha = \sqrt{7}$ into this expression.

The expression becomes:

We need to find the positive square root of $8 + 2\sqrt{7}$. This looks like a form that can be simplified if it is a perfect square of a binomial involving square roots, i.e., in the form $(\sqrt{A} + \sqrt{B})^2$ or $(\sqrt{A} - \sqrt{B})^2$.

Using the Form \( \sqrt{a + 2\sqrt{b}} \)

The expression $8 + 2\sqrt{7}$ is already in the form $a + 2\sqrt{b}$, where $a=8$ and $b=7$. We look for two numbers, let's call them A and B, such that their sum is the number outside the square root (8) and their product is the number inside the inner square root (7).

• Sum $A + B = 8$
• Product $AB = 7$

We need to find two numbers whose sum is 8 and product is 7. By thinking about the factors of 7, which are 1 and 7, we can see that $1 + 7 = 8$ and $1 \times 7 = 7$. So, the two numbers are 7 and 1.

Using these numbers, we can rewrite the expression $8 + 2\sqrt{7}$ as:

This is in the form $A + B + 2\sqrt{AB}$, which is the expansion of $(\sqrt{A} + \sqrt{B})^2$.

So, $8 + 2\sqrt{7} = (\sqrt{7} + 1)^2$.

Finding the Final Positive Square Root

We need the positive square root of this expression:

Since $\sqrt{7}$ is positive and 1 is positive, their sum $\sqrt{7} + 1$ is positive. The positive square root of a positive number squared is the number itself.

Therefore, the positive square root of $8 + 2\alpha$ is $\sqrt{7} + 1$.

Comparing the Result with Options

Let's compare our calculated result with the given options:

• Option 1: $2 + \sqrt{3}$
• Option 2: $3 - \sqrt{2}$
• Option 3: $\sqrt{7} - 1$
• Option 4: $\sqrt{7} + 1$

Our result, $\sqrt{7} + 1$, matches Option 4.

Revision Table: Key Concepts Used

Additional Information: Simplifying Square Roots of the Form \( \sqrt{a \pm \sqrt{b}} \)

Expressions like $\sqrt{a + \sqrt{b}}$ or $\sqrt{a - \sqrt{b}}$ can sometimes be simplified into the form $\sqrt{x} + \sqrt{y}$ or $\sqrt{x} - \sqrt{y}$. The key is to manipulate the expression inside the square root to match the expansion of $(\sqrt{x} + \sqrt{y})^2 = x + y + 2\sqrt{xy}$ or $(\sqrt{x} - \sqrt{y})^2 = x + y - 2\sqrt{xy}$.

The expression we encountered was $8 + 2\sqrt{7}$. This is already in the form $a + 2\sqrt{b}$, which directly matches the structure $x+y + 2\sqrt{xy}$. So, we looked for $x$ and $y$ such that:

• $x + y = 8$ (the number outside the $2\sqrt{\cdot}$)
• $xy = 7$ (the number inside the inner square root)

Once we find such $x$ and $y$, the simplified form is $\sqrt{x} + \sqrt{y}$ (if the original sign is +) or $\sqrt{x} - \sqrt{y}$ (if the original sign is -). In our case, we found $x=7$ and $y=1$ (or vice versa), and the sign was +, so the result is $\sqrt{7} + \sqrt{1} = \sqrt{7} + 1$. It's important that $x$ and $y$ are positive to ensure the square roots are real numbers and the simplification is valid.