Which Expression Is Equivalent To 7 A 3 ( 6 A 2 + A ) 2 − 4 A 6 7a^3(6a^2+a)^2 - 4a^6 7 A 3 ( 6 A 2 + A ) 2 − 4 A 6 ?A. 252 A 7 + 84 A 6 + 7 A 5 252a^7 + 84a^6 + 7a^5 252 A 7 + 84 A 6 + 7 A 5 B. 252 A 7 + 80 A 6 + 7 A 5 252a^7 + 80a^6 + 7a^5 252 A 7 + 80 A 6 + 7 A 5 C. 252 A 7 − 4 A 6 + 7 A 5 252a^7 - 4a^6 + 7a^5 252 A 7 − 4 A 6 + 7 A 5 D. 252 A 7 − 4 A 6 + A 2 252a^7 - 4a^6 + A^2 252 A 7 − 4 A 6 + A 2

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Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is an essential skill for any math enthusiast. In this article, we will explore the process of simplifying algebraic expressions, with a focus on the given expression 7a3(6a2+a)2−4a67a^3(6a^2+a)^2 - 4a^6. We will break down the expression into manageable parts, apply the necessary algebraic rules, and arrive at the equivalent expression.

Understanding the Given Expression

The given expression is 7a3(6a2+a)2−4a67a^3(6a^2+a)^2 - 4a^6. To simplify this expression, we need to understand the rules of exponents and how to expand algebraic expressions.

Expanding the Expression

The expression (6a2+a)2(6a^2+a)^2 can be expanded using the formula (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2. Applying this formula, we get:

(6a2+a)2=(6a2)2+2(6a2)(a)+a2(6a^2+a)^2 = (6a^2)^2 + 2(6a^2)(a) + a^2

=36a4+12a3+a2= 36a^4 + 12a^3 + a^2

Now, we can substitute this expanded expression back into the original expression:

7a3(36a4+12a3+a2)−4a67a^3(36a^4 + 12a^3 + a^2) - 4a^6

Distributing the Terms

To simplify the expression further, we need to distribute the terms inside the parentheses. This means multiplying each term inside the parentheses by the factor outside the parentheses, which is 7a37a^3.

7a3(36a4)+7a3(12a3)+7a3(a2)−4a67a^3(36a^4) + 7a^3(12a^3) + 7a^3(a^2) - 4a^6

=252a7+84a6+7a5−4a6= 252a^7 + 84a^6 + 7a^5 - 4a^6

Combining Like Terms

Now that we have distributed the terms, we can combine like terms. In this case, we have two terms with the same exponent, a6a^6. We can combine these terms by adding their coefficients.

252a7+(84a6−4a6)+7a5252a^7 + (84a^6 - 4a^6) + 7a^5

=252a7+80a6+7a5= 252a^7 + 80a^6 + 7a^5

Conclusion

In conclusion, the expression 7a3(6a2+a)2−4a67a^3(6a^2+a)^2 - 4a^6 is equivalent to 252a7+80a6+7a5252a^7 + 80a^6 + 7a^5. This expression can be obtained by expanding the given expression, distributing the terms, and combining like terms.

Answer

The correct answer is:

  • A. 252a7+84a6+7a5252a^7 + 84a^6 + 7a^5 (Incorrect)
  • B. 252a7+80a6+7a5252a^7 + 80a^6 + 7a^5 (Correct)
  • C. 252a7−4a6+7a5252a^7 - 4a^6 + 7a^5 (Incorrect)
  • D. 252a7−4a6+a2252a^7 - 4a^6 + a^2 (Incorrect)

Final Thoughts

Q: What is the first step in simplifying an algebraic expression?

A: The first step in simplifying an algebraic expression is to understand the expression and identify any like terms. Like terms are terms that have the same variable and exponent.

Q: How do I expand an algebraic expression?

A: To expand an algebraic expression, you can use the distributive property, which states that a(b + c) = ab + ac. You can also use the formula (a + b)^2 = a^2 + 2ab + b^2 to expand expressions of the form (a + b)^2.

Q: What is the difference between a like term and a unlike term?

A: A like term is a term that has the same variable and exponent. For example, 2x^2 and 3x^2 are like terms because they both have the variable x and the exponent 2. A unlike term is a term that has a different variable or exponent. For example, 2x^2 and 3y^2 are unlike terms because they have different variables (x and y) and exponents (2 and 2).

Q: How do I combine like terms?

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

Q: What is the order of operations in simplifying an algebraic expression?

A: The order of operations in simplifying an algebraic expression is:

  1. Evaluate any expressions inside parentheses.
  2. Exponentiate any terms with exponents.
  3. Multiply any terms.
  4. Add or subtract any terms.

Q: How do I simplify an expression with a negative exponent?

A: To simplify an expression with a negative exponent, you can rewrite the expression with a positive exponent by taking the reciprocal of the term. For example, if you have the expression 2x^-2, you can rewrite it as 1/(2x^2).

Q: What is the difference between a simplified expression and a factored expression?

A: A simplified expression is an expression that has been reduced to its simplest form by combining like terms and eliminating any unnecessary parentheses. A factored expression is an expression that has been written as a product of simpler expressions. For example, the expression 2x^2 + 4x + 2 can be simplified to 2(x^2 + 2x + 1), but it can also be factored as 2(x + 1)(x + 1).

Q: How do I determine if an expression is in its simplest form?

A: To determine if an expression is in its simplest form, you can check the following:

  • Are there any like terms that can be combined?
  • Are there any unnecessary parentheses that can be eliminated?
  • Are there any terms that can be simplified by taking the reciprocal of a term with a negative exponent?

If the answer to any of these questions is yes, then the expression is not in its simplest form.

Conclusion

Simplifying algebraic expressions is an essential skill for any math enthusiast. By understanding the rules of exponents and how to expand algebraic expressions, you can simplify even the most complex expressions and arrive at the equivalent expression. Remember to follow the order of operations, combine like terms, and eliminate any unnecessary parentheses to ensure that your expression is in its simplest form.