Simplify The Expression: \[$(a+8)(7a+7)\$\]

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Introduction

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In algebra, simplifying expressions is a crucial skill that helps us solve equations and manipulate mathematical statements. The given expression, $(a+8)(7a+7)$, is a product of two binomials, and we need to simplify it using the distributive property. In this article, we will walk through the step-by-step process of simplifying the expression and provide a clear understanding of the underlying concepts.

Understanding the Distributive Property

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The distributive property is a fundamental concept in algebra that allows us to expand a product of two or more binomials. It states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$$a(b+c) = ab + ac$$

This property can be extended to more than two binomials, and it is a powerful tool for simplifying expressions.

Applying the Distributive Property

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To simplify the expression $(a+8)(7a+7)$, we will apply the distributive property by multiplying each term in the first binomial with each term in the second binomial.

$$(a+8)(7a+7) = a(7a+7) + 8(7a+7)$$

Expanding the Terms

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Now, we will expand each term by multiplying the corresponding terms.

$$a(7a+7) = 7a^2 + 7a$$

$$8(7a+7) = 56a + 56$$

Combining Like Terms

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The next step is to combine like terms, which involves adding or subtracting terms with the same variable and exponent.

$$7a^2 + 7a + 56a + 56 = 7a^2 + 63a + 56$$

Final Simplified Expression

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After applying the distributive property, expanding the terms, and combining like terms, we arrive at the final simplified expression:

$$(a+8)(7a+7) = 7a^2 + 63a + 56$$

Conclusion

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Simplifying the expression $(a+8)(7a+7)$ using the distributive property involves applying the concept to a product of two binomials. By expanding the terms and combining like terms, we arrive at the final simplified expression. This process demonstrates the importance of understanding and applying the distributive property in algebra.

Real-World Applications

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The distributive property has numerous real-world applications in various fields, including:

• Physics: When calculating the force exerted on an object by multiple forces, the distributive property is used to simplify the expression.
• Engineering: In designing complex systems, the distributive property is applied to simplify expressions and make calculations more manageable.
• Computer Science: In programming, the distributive property is used to simplify expressions and optimize code.

Tips and Tricks

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When simplifying expressions using the distributive property, remember to:

• Apply the property systematically: Multiply each term in the first binomial with each term in the second binomial.
• Combine like terms carefully: Add or subtract terms with the same variable and exponent.
• Check your work: Verify that the final simplified expression is correct by plugging in values or using a calculator.

By following these tips and tricks, you will become more confident and proficient in simplifying expressions using the distributive property.

Common Mistakes to Avoid

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When simplifying expressions using the distributive property, be aware of the following common mistakes:

• Forgetting to apply the property: Failing to multiply each term in the first binomial with each term in the second binomial.
• Incorrectly combining like terms: Adding or subtracting terms with different variables or exponents.
• Not checking your work: Failing to verify that the final simplified expression is correct.

By avoiding these common mistakes, you will ensure that your simplifications are accurate and reliable.

Practice Problems

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To reinforce your understanding of the distributive property, try simplifying the following expressions:

• $(x+3)(4x+2)$
• $(2y-1)(3y+5)$
• $(a-2)(b+4)$

By practicing these problems, you will become more comfortable and confident in applying the distributive property to simplify expressions.

Conclusion

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Simplifying the expression $(a+8)(7a+7)$ using the distributive property involves applying the concept to a product of two binomials. By expanding the terms and combining like terms, we arrive at the final simplified expression. This process demonstrates the importance of understanding and applying the distributive property in algebra. With practice and patience, you will become proficient in simplifying expressions using the distributive property and apply it to real-world problems.

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Q: What is the distributive property, and how is it used in algebra?

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A: The distributive property is a fundamental concept in algebra that allows us to expand a product of two or more binomials. It states that for any real numbers $a$, $b$, and $c$, the following equation holds:

$$a(b+c) = ab + ac$$

This property can be extended to more than two binomials, and it is a powerful tool for simplifying expressions.

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

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A: To simplify an expression using the distributive property, follow these steps:

1. Apply the property systematically: Multiply each term in the first binomial with each term in the second binomial.
2. Combine like terms carefully: Add or subtract terms with the same variable and exponent.
3. Check your work: Verify that the final simplified expression is correct by plugging in values or using a calculator.

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

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A: When simplifying expressions using the distributive property, be aware of the following common mistakes:

• Forgetting to apply the property: Failing to multiply each term in the first binomial with each term in the second binomial.
• Incorrectly combining like terms: Adding or subtracting terms with different variables or exponents.
• Not checking your work: Failing to verify that the final simplified expression is correct.

Q: How do I simplify an expression with multiple binomials?

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A: To simplify an expression with multiple binomials, apply the distributive property systematically, multiplying each term in the first binomial with each term in the second binomial, and then combining like terms.

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

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A: Yes, the distributive property can be used to simplify expressions with variables and constants. When multiplying a variable with a constant, the constant is distributed to each term in the binomial.

Q: How do I check my work when simplifying expressions with the distributive property?

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A: To check your work, plug in values or use a calculator to verify that the final simplified expression is correct. You can also use a graphing calculator or a computer algebra system to check your work.

Q: What are some real-world applications of the distributive property?

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A: The distributive property has numerous real-world applications in various fields, including:

• Physics: When calculating the force exerted on an object by multiple forces, the distributive property is used to simplify the expression.
• Engineering: In designing complex systems, the distributive property is applied to simplify expressions and make calculations more manageable.
• Computer Science: In programming, the distributive property is used to simplify expressions and optimize code.

Q: How can I practice simplifying expressions with the distributive property?

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A: To practice simplifying expressions with the distributive property, try the following:

• Solve practice problems: Try simplifying expressions with the distributive property using different variables and binomials.
• Use online resources: Utilize online resources, such as algebra worksheets and practice problems, to practice simplifying expressions with the distributive property.
• Work with a tutor or teacher: Collaborate with a tutor or teacher to practice simplifying expressions with the distributive property and receive feedback on your work.

By following these tips and practicing regularly, you will become more confident and proficient in simplifying expressions using the distributive property.