Find The Product.\[$\left(3a^4 + 4\right)^2\$\]A. \[$6a^4 + 8\$\]B. \[$9a^8 + 16\$\]C. \[$9a^8 + 24a^4 + 16\$\]D. \[$9a^{16} + 24a^4 + 16\$\]

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Introduction

In algebra, expanding the square of a binomial expression is a crucial skill that helps us simplify complex expressions and solve equations. In this article, we will focus on finding the product of the given expression (3a4+4)2\left(3a^4 + 4\right)^2. We will use the formula for expanding the square of a binomial expression and apply it step by step to arrive at the correct answer.

The Formula for Expanding the Square of a Binomial Expression

The formula for expanding the square of a binomial expression is:

(a+b)2=a2+2ab+b2\left(a + b\right)^2 = a^2 + 2ab + b^2

This formula can be applied to any binomial expression of the form (a+b)2\left(a + b\right)^2, where aa and bb are any algebraic expressions.

Applying the Formula to the Given Expression

Now, let's apply the formula to the given expression (3a4+4)2\left(3a^4 + 4\right)^2. We can rewrite the expression as:

(3a4+4)2=(3a4)2+2(3a4)(4)+(4)2\left(3a^4 + 4\right)^2 = \left(3a^4\right)^2 + 2\left(3a^4\right)\left(4\right) + \left(4\right)^2

Expanding the Terms

Now, let's expand the terms in the expression:

(3a4)2=(3)2(a4)2=9a8\left(3a^4\right)^2 = \left(3\right)^2\left(a^4\right)^2 = 9a^8

2(3a4)(4)=2(3)(4)a4=24a42\left(3a^4\right)\left(4\right) = 2\left(3\right)\left(4\right)a^4 = 24a^4

(4)2=16\left(4\right)^2 = 16

Combining the Terms

Now, let's combine the terms in the expression:

9a8+24a4+169a^8 + 24a^4 + 16

Conclusion

In conclusion, the product of the given expression (3a4+4)2\left(3a^4 + 4\right)^2 is 9a8+24a4+169a^8 + 24a^4 + 16. This is the correct answer, and it can be verified by plugging in values for aa and checking the result.

Comparison with the Options

Now, let's compare the correct answer with the options:

A. 6a4+86a^4 + 8

B. 9a8+169a^8 + 16

C. 9a8+24a4+169a^8 + 24a^4 + 16

D. 9a16+24a4+169a^{16} + 24a^4 + 16

The correct answer is option C, which is 9a8+24a4+169a^8 + 24a^4 + 16.

Tips and Tricks

Here are some tips and tricks for expanding the square of a binomial expression:

  • Use the formula (a+b)2=a2+2ab+b2\left(a + b\right)^2 = a^2 + 2ab + b^2 to expand the square of a binomial expression.
  • Apply the formula to any binomial expression of the form (a+b)2\left(a + b\right)^2, where aa and bb are any algebraic expressions.
  • Expand the terms in the expression by multiplying the coefficients and variables.
  • Combine the terms in the expression by adding or subtracting like terms.
  • Check the result by plugging in values for the variables and checking the result.

Practice Problems

Here are some practice problems for expanding the square of a binomial expression:

  • Find the product of the expression (2x+3)2\left(2x + 3\right)^2.
  • Find the product of the expression (x2+2)2\left(x^2 + 2\right)^2.
  • Find the product of the expression (3x2+4)2\left(3x^2 + 4\right)^2.

Conclusion

In conclusion, expanding the square of a binomial expression is a crucial skill that helps us simplify complex expressions and solve equations. By applying the formula (a+b)2=a2+2ab+b2\left(a + b\right)^2 = a^2 + 2ab + b^2 and following the steps outlined in this article, we can find the product of any binomial expression of the form (a+b)2\left(a + b\right)^2. We can also use the tips and tricks outlined in this article to make the process easier and more efficient.

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Introduction

In our previous article, we discussed how to expand the square of a binomial expression using the formula (a+b)2=a2+2ab+b2\left(a + b\right)^2 = a^2 + 2ab + b^2. In this article, we will provide a Q&A guide to help you understand the concept better and apply it to different types of problems.

Q: What is the formula for expanding the square of a binomial expression?

A: The formula for expanding the square of a binomial expression is (a+b)2=a2+2ab+b2\left(a + b\right)^2 = a^2 + 2ab + b^2.

Q: How do I apply the formula to a binomial expression?

A: To apply the formula, you need to identify the values of aa and bb in the binomial expression. Then, you can plug these values into the formula and expand the expression.

Q: What if the binomial expression has variables with exponents?

A: If the binomial expression has variables with exponents, you need to apply the exponent rule when expanding the expression. For example, if the expression is (2x3+3)2\left(2x^3 + 3\right)^2, you would expand it as follows:

(2x3+3)2=(2x3)2+2(2x3)(3)+(3)2\left(2x^3 + 3\right)^2 = \left(2x^3\right)^2 + 2\left(2x^3\right)\left(3\right) + \left(3\right)^2

Q: How do I handle negative coefficients in a binomial expression?

A: When a binomial expression has a negative coefficient, you can handle it by distributing the negative sign to all the terms in the expression. For example, if the expression is (−2x+3)2\left(-2x + 3\right)^2, you would expand it as follows:

(−2x+3)2=(−2x)2+2(−2x)(3)+(3)2\left(-2x + 3\right)^2 = \left(-2x\right)^2 + 2\left(-2x\right)\left(3\right) + \left(3\right)^2

Q: Can I use the formula to expand expressions with more than two terms?

A: No, the formula is only applicable to binomial expressions with two terms. If you have an expression with more than two terms, you need to use other methods to expand it.

Q: How do I check my answer when expanding a binomial expression?

A: To check your answer, you can plug in values for the variables and check if the result is correct. You can also use a calculator or a computer algebra system to verify your answer.

Q: What are some common mistakes to avoid when expanding binomial expressions?

A: Some common mistakes to avoid when expanding binomial expressions include:

  • Forgetting to distribute the negative sign to all the terms in the expression.
  • Not applying the exponent rule when expanding expressions with variables with exponents.
  • Not checking the result by plugging in values for the variables.

Q: How can I practice expanding binomial expressions?

A: You can practice expanding binomial expressions by working through examples and exercises in a textbook or online resource. You can also try creating your own problems and solving them to test your understanding of the concept.

Conclusion

In conclusion, expanding the square of a binomial expression is a crucial skill that helps us simplify complex expressions and solve equations. By applying the formula (a+b)2=a2+2ab+b2\left(a + b\right)^2 = a^2 + 2ab + b^2 and following the steps outlined in this article, we can expand binomial expressions with ease. We can also use the tips and tricks outlined in this article to make the process easier and more efficient.