The Product { (7-\sqrt{10})(7-\sqrt{10})$}$ In Simplest Form.

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems more efficiently. One of the most common ways to simplify an expression is by using the distributive property, which states that for any real numbers a, b, and c, a(b + c) = ab + ac. In this article, we will use the distributive property to simplify the product (7−10)(7−10)(7-\sqrt{10})(7-\sqrt{10}).

Understanding the Expression

The given expression is a product of two binomials, (7−10)(7-\sqrt{10}) and (7−10)(7-\sqrt{10}). To simplify this expression, we need to multiply the two binomials using the distributive property.

Applying the Distributive Property

The distributive property states that for any real numbers a, b, and c, a(b + c) = ab + ac. In this case, we have:

(7−10)(7−10)=(7)(7)−(7)(10)−(10)(7)+(10)(10)(7-\sqrt{10})(7-\sqrt{10}) = (7)(7) - (7)(\sqrt{10}) - (\sqrt{10})(7) + (\sqrt{10})(\sqrt{10})

Simplifying the Expression

Now, let's simplify the expression by combining like terms.

(7)(7)=49(7)(7) = 49

−(7)(10)=−710-(7)(\sqrt{10}) = -7\sqrt{10}

−(10)(7)=−710-(\sqrt{10})(7) = -7\sqrt{10}

(10)(10)=10(\sqrt{10})(\sqrt{10}) = 10

Combining Like Terms

Now, let's combine the like terms.

49−710−710+1049 - 7\sqrt{10} - 7\sqrt{10} + 10

Final Simplification

Now, let's simplify the expression further by combining the like terms.

49−1410+1049 - 14\sqrt{10} + 10

=59−1410= 59 - 14\sqrt{10}

Conclusion

In this article, we used the distributive property to simplify the product (7−10)(7−10)(7-\sqrt{10})(7-\sqrt{10}). We first applied the distributive property to multiply the two binomials, and then simplified the expression by combining like terms. The final simplified form of the expression is 59−141059 - 14\sqrt{10}.

Frequently Asked Questions

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any real numbers a, b, and c, a(b + c) = ab + ac.

Q: How do I simplify an expression using the distributive property?

A: To simplify an expression using the distributive property, you need to multiply the terms inside the parentheses using the distributive property, and then combine like terms.

Q: What is the final simplified form of the expression (7−10)(7−10)(7-\sqrt{10})(7-\sqrt{10})?

A: The final simplified form of the expression (7−10)(7−10)(7-\sqrt{10})(7-\sqrt{10}) is 59−141059 - 14\sqrt{10}.

Further Reading

If you want to learn more about simplifying expressions using the distributive property, you can check out the following resources:

  • Mathway: A online math problem solver that can help you simplify expressions.
  • Khan Academy: A free online learning platform that offers video lessons on math and other subjects.
  • Wolfram Alpha: A online calculator that can help you simplify expressions and solve math problems.

References

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Introduction

In our previous article, we discussed how to simplify the product (7−10)(7−10)(7-\sqrt{10})(7-\sqrt{10}) using the distributive property. In this article, we will answer some frequently asked questions about simplifying expressions using the distributive property.

Q&A

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any real numbers a, b, and c, a(b + c) = ab + ac. This means that you can multiply a single term to two or more terms inside the parentheses.

Q: How do I simplify an expression using the distributive property?

A: To simplify an expression using the distributive property, you need to multiply the terms inside the parentheses using the distributive property, and then combine like terms. Here's a step-by-step guide:

  1. Multiply the terms inside the parentheses using the distributive property.
  2. Combine like terms by adding or subtracting the coefficients of the same variables.
  3. Simplify the expression by combining the like terms.

Q: What are like terms?

A: Like terms are terms that have the same variable raised to the same power. For example, 2x and 3x are like terms because they both have the variable x raised to the power of 1.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the coefficients of the same variables. For example, if you have 2x + 3x, you can combine the like terms by adding the coefficients: 2x + 3x = 5x.

Q: What is the final simplified form of the expression (7−10)(7−10)(7-\sqrt{10})(7-\sqrt{10})?

A: The final simplified form of the expression (7−10)(7−10)(7-\sqrt{10})(7-\sqrt{10}) is 59−141059 - 14\sqrt{10}.

Q: Can I use the distributive property to simplify expressions with variables?

A: Yes, you can use the distributive property to simplify expressions with variables. For example, if you have (x + 2)(x + 3), you can multiply the terms inside the parentheses using the distributive property and then combine like terms.

Q: What are some common mistakes to avoid when simplifying expressions using the distributive property?

A: Some common mistakes to avoid when simplifying expressions using the distributive property include:

  • Forgetting to multiply the terms inside the parentheses.
  • Not combining like terms.
  • Making errors when multiplying or combining like terms.

Tips and Tricks

Tip 1: Use the distributive property to simplify expressions with variables.

The distributive property can be used to simplify expressions with variables. For example, if you have (x + 2)(x + 3), you can multiply the terms inside the parentheses using the distributive property and then combine like terms.

Tip 2: Combine like terms carefully.

When combining like terms, make sure to add or subtract the coefficients of the same variables carefully. For example, if you have 2x + 3x, you can combine the like terms by adding the coefficients: 2x + 3x = 5x.

Tip 3: Check your work.

When simplifying expressions using the distributive property, make sure to check your work carefully. This will help you avoid making errors and ensure that your final answer is correct.

Conclusion

In this article, we answered some frequently asked questions about simplifying expressions using the distributive property. We also provided some tips and tricks to help you simplify expressions using the distributive property. By following these tips and tricks, you can simplify expressions with confidence and accuracy.

Further Reading

If you want to learn more about simplifying expressions using the distributive property, you can check out the following resources:

  • Mathway: A online math problem solver that can help you simplify expressions.
  • Khan Academy: A free online learning platform that offers video lessons on math and other subjects.
  • Wolfram Alpha: A online calculator that can help you simplify expressions and solve math problems.

References