Simplify The Expression: ${ (\sqrt{8}+2)(\sqrt{8}-1)+\frac{5 \sqrt{6}}{\sqrt{3}} \times 2 }$

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Introduction

In this article, we will simplify the given expression using basic algebraic and trigonometric concepts. The expression involves the addition and multiplication of square roots, as well as the multiplication of a fraction by a constant. Our goal is to simplify the expression to its simplest form.

The Given Expression

The given expression is:

(8+2)(8−1)+563×2(\sqrt{8}+2)(\sqrt{8}-1)+\frac{5 \sqrt{6}}{\sqrt{3}} \times 2

Step 1: Simplify the Square Roots

The first step is to simplify the square roots in the expression. We can rewrite 8\sqrt{8} as 4×2\sqrt{4 \times 2}, which simplifies to 222\sqrt{2}.

import math

sqrt_8 = math.sqrt(8)



simplified_sqrt_8 = 2 * math.sqrt(2)

Step 2: Apply the Difference of Squares Formula

The next step is to apply the difference of squares formula to the first part of the expression. The difference of squares formula states that (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2. In this case, we have:

(8+2)(8−1)=(8)2−(1)2(\sqrt{8}+2)(\sqrt{8}-1) = (\sqrt{8})^2 - (1)^2

# Apply the difference of squares formula
result = (simplified_sqrt_8)**2 - 1**2

Step 3: Simplify the Result

The result of the difference of squares formula is:

8−1=78 - 1 = 7

Step 4: Simplify the Fraction

The next step is to simplify the fraction 563\frac{5 \sqrt{6}}{\sqrt{3}}. We can rewrite this fraction as:

563=52×33\frac{5 \sqrt{6}}{\sqrt{3}} = \frac{5 \sqrt{2} \times \sqrt{3}}{\sqrt{3}}

# Simplify the fraction
simplified_fraction = (5 * math.sqrt(2) * math.sqrt(3)) / math.sqrt(3)

Step 5: Cancel Out the Common Factors

The final step is to cancel out the common factors in the fraction. In this case, we can cancel out the 3\sqrt{3} in the numerator and denominator:

52×33=52\frac{5 \sqrt{2} \times \sqrt{3}}{\sqrt{3}} = 5 \sqrt{2}

Step 6: Multiply the Simplified Fraction by 2

The final step is to multiply the simplified fraction by 2:

52×2=1025 \sqrt{2} \times 2 = 10 \sqrt{2}

Conclusion

In conclusion, we have simplified the given expression using basic algebraic and trigonometric concepts. The final simplified expression is:

7+1027 + 10 \sqrt{2}

This expression cannot be simplified further, and it is the simplest form of the original expression.

Final Answer

The final answer is:

7+1027 + 10 \sqrt{2}

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Introduction

In our previous article, we simplified the given expression using basic algebraic and trigonometric concepts. In this article, we will answer some frequently asked questions related to the simplification of the expression.

Q&A

Q: What is the difference of squares formula?

A: The difference of squares formula is a mathematical formula that states that (a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2. This formula can be used to simplify expressions that involve the product of two binomials.

Q: How do I apply the difference of squares formula?

A: To apply the difference of squares formula, you need to identify the two binomials in the expression and multiply them together. Then, you can simplify the result using the formula.

Q: What is the simplified form of the expression (8+2)(8−1)(\sqrt{8}+2)(\sqrt{8}-1)?

A: The simplified form of the expression (8+2)(8−1)(\sqrt{8}+2)(\sqrt{8}-1) is 77.

Q: How do I simplify the fraction 563\frac{5 \sqrt{6}}{\sqrt{3}}?

A: To simplify the fraction 563\frac{5 \sqrt{6}}{\sqrt{3}}, you need to rewrite it as 52×33\frac{5 \sqrt{2} \times \sqrt{3}}{\sqrt{3}} and then cancel out the common factors.

Q: What is the final simplified expression?

A: The final simplified expression is 7+1027 + 10 \sqrt{2}.

Q: Can the expression be simplified further?

A: No, the expression cannot be simplified further.

Q: What is the final answer?

A: The final answer is 7+1027 + 10 \sqrt{2}.

Common Mistakes

Mistake 1: Not Simplifying the Square Roots

A common mistake when simplifying the expression is not simplifying the square roots. Make sure to simplify the square roots before applying the difference of squares formula.

Mistake 2: Not Canceling Out the Common Factors

Another common mistake is not canceling out the common factors in the fraction. Make sure to cancel out the common factors before simplifying the expression.

Mistake 3: Not Multiplying the Simplified Fraction by 2

A final common mistake is not multiplying the simplified fraction by 2. Make sure to multiply the simplified fraction by 2 before simplifying the expression.

Conclusion

In conclusion, we have answered some frequently asked questions related to the simplification of the expression. We have also discussed some common mistakes that can occur when simplifying the expression. By following the steps outlined in this article, you should be able to simplify the expression correctly.

Final Answer

The final answer is:

7+1027 + 10 \sqrt{2}