What Is The Following Product?\[$(x \sqrt{7} - 3 \sqrt{8})(x \sqrt{7} - 3 \sqrt{8})\$\]A. \[$7x^2 + 72\$\]B. \[$7x^2 - 12x \sqrt{14} + 72\$\]C. \[$7x^2 - 12x \sqrt{14} - 72\$\]D. \[$7x^2 - 72\$\]

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Understanding the Problem

The given problem involves finding the product of two expressions, each containing a square root term. To solve this, we need to apply the rules of algebra and simplify the expression step by step.

Step 1: Expand the Expression

The given expression is {(x \sqrt{7} - 3 \sqrt{8})(x \sqrt{7} - 3 \sqrt{8})$}$. To expand this, we need to apply the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac.

Using this property, we can expand the given expression as follows:

{(x \sqrt{7} - 3 \sqrt{8})(x \sqrt{7} - 3 \sqrt{8}) = x^2 \sqrt{7} \sqrt{7} - 3x \sqrt{7} \sqrt{8} - 3x \sqrt{8} \sqrt{7} + 9 \sqrt{8} \sqrt{8}$

Step 2: Simplify the Expression

Now, we can simplify the expression by combining like terms and applying the rules of algebra.

First, we can simplify the square root terms:

[77=7$\[\sqrt{7} \sqrt{7} = 7\$ \[\sqrt{8} \sqrt{8} = 8$

Next, we can combine the like terms:

[$x^2 \sqrt{7} \sqrt{7} - 3x \sqrt{7} \sqrt{8} - 3x \sqrt{8} \sqrt{7} + 9 \sqrt{8} \sqrt{8} = x^2 (7) - 3x (\sqrt{7} \sqrt{8}) - 3x (\sqrt{8} \sqrt{7}) + 9 (8)$

Now, we can simplify the expression further by combining the like terms:

[$x^2 (7) - 3x (\sqrt{7} \sqrt{8}) - 3x (\sqrt{8} \sqrt{7}) + 9 (8) = 7x^2 - 3x (\sqrt{7} \sqrt{8}) - 3x (\sqrt{8} \sqrt{7}) + 72$

Step 3: Final Simplification

Now, we can simplify the expression further by combining the like terms:

[$7x^2 - 3x (\sqrt{7} \sqrt{8}) - 3x (\sqrt{8} \sqrt{7}) + 72 = 7x^2 - 3x (\sqrt{7} \sqrt{8} + \sqrt{8} \sqrt{7}) + 72$

Using the commutative property of multiplication, we can rewrite the expression as:

[$7x^2 - 3x (\sqrt{7} \sqrt{8} + \sqrt{8} \sqrt{7}) + 72 = 7x^2 - 3x (2 \sqrt{7} \sqrt{8}) + 72$

Now, we can simplify the expression further by combining the like terms:

[$7x^2 - 3x (2 \sqrt{7} \sqrt{8}) + 72 = 7x^2 - 6x \sqrt{7} \sqrt{8} + 72$

Using the fact that [$\sqrt{7} \sqrt{8} = \sqrt{56} = \sqrt{4 \cdot 14} = 2 \sqrt{14}$, we can rewrite the expression as:

[$7x^2 - 6x \sqrt{7} \sqrt{8} + 72 = 7x^2 - 6x (2 \sqrt{14}) + 72$

Now, we can simplify the expression further by combining the like terms:

[$7x^2 - 6x (2 \sqrt{14}) + 72 = 7x^2 - 12x \sqrt{14} + 72$

Conclusion

Therefore, the final answer is:

[7x^2 - 12x \sqrt{14} + 72}

This is option B.

Answer Key

A. ${7x^2 + 72} B. ${7x^2 - 12x \sqrt{14} + 72} C. ${7x^2 - 12x \sqrt{14} - 72} D. ${7x^2 - 72}

Q: What is the product of two expressions, each containing a square root term? A: The product of two expressions, each containing a square root term, can be found by applying the distributive property and simplifying the resulting expression.

Q: How do I apply the distributive property? A: To apply the distributive property, you need to multiply each term in the first expression by each term in the second expression.

Q: What is the distributive property? A: The distributive property is a rule of algebra that states that for any real numbers a, b, and c, a(b + c) = ab + ac.

Q: How do I simplify the expression? A: To simplify the expression, you need to combine like terms and apply the rules of algebra.

Q: What are like terms? A: Like terms are terms that have the same variable raised to the same power.

Q: How do I combine like terms? A: To combine like terms, you need to add or subtract the coefficients of the like terms.

Q: What is the coefficient of a term? A: The coefficient of a term is the number that is multiplied by the variable.

Q: How do I apply the rules of algebra? A: To apply the rules of algebra, you need to follow the order of operations (PEMDAS) and simplify the expression step by step.

Q: What is PEMDAS? A: PEMDAS is a mnemonic device that stands for "Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction". It is used to remember the order of operations.

Q: How do I simplify the expression further? A: To simplify the expression further, you need to combine the like terms and apply the rules of algebra.

Q: What is the final answer? A: The final answer is ${7x^2 - 12x \sqrt{14} + 72}.

Q: What is the correct answer? A: The correct answer is option B.

Q: What are the other options? A: The other options are:

A. ${7x^2 + 72} C. ${7x^2 - 12x \sqrt{14} - 72} D. ${7x^2 - 72}

Q: How do I choose the correct answer? A: To choose the correct answer, you need to compare the final answer with the options and select the one that matches.

Q: What is the importance of simplifying expressions? A: Simplifying expressions is important because it helps to make the expression easier to understand and work with.

Q: How do I apply simplification in real-life situations? A: Simplification is applied in real-life situations by breaking down complex problems into simpler ones and solving them step by step.

Q: What are some common applications of simplification? A: Some common applications of simplification include:

  • Algebra
  • Geometry
  • Trigonometry
  • Calculus
  • Physics
  • Engineering

Q: How do I practice simplification? A: To practice simplification, you can try solving problems that involve simplifying expressions and equations.

Q: What are some resources for learning simplification? A: Some resources for learning simplification include:

  • Textbooks
  • Online tutorials
  • Video lectures
  • Practice problems
  • Study groups