Which Expressions Are Equivalent To $25x^4 - 64$? Select Three Options.A. $25x + 40x - 40x - 64$B. $25x + 13x - 13x - 64$C. $(5x^2 + 8)(5x^2 - 8$\]D. $(x^2 + 13)(x^2 - 13$\]E. $(5x^2 - 8)^2$

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In mathematics, equivalent expressions are those that have the same value for all possible values of the variable. In this article, we will explore which expressions are equivalent to the given expression $25x^4 - 64$. We will examine each option carefully and determine whether it is equivalent to the given expression.

Option A: $25x + 40x - 40x - 64$

Let's start by analyzing option A. This expression is a combination of several terms, but it is not immediately clear whether it is equivalent to the given expression. To determine whether it is equivalent, we need to simplify the expression.

$25x + 40x - 40x - 64$

We can start by combining like terms:

$65x - 64$

However, this expression is not equivalent to the given expression $25x^4 - 64$. The reason is that the exponent of $x$ is 1, whereas the exponent of $x$ in the given expression is 4. Therefore, option A is not equivalent to the given expression.

Option B: $25x + 13x - 13x - 64$

Next, let's analyze option B. This expression is also a combination of several terms, but it is not immediately clear whether it is equivalent to the given expression. To determine whether it is equivalent, we need to simplify the expression.

$25x + 13x - 13x - 64$

We can start by combining like terms:

$25x - 64$

However, this expression is not equivalent to the given expression $25x^4 - 64$. The reason is that the exponent of $x$ is 1, whereas the exponent of $x$ in the given expression is 4. Therefore, option B is not equivalent to the given expression.

Option C: $(5x^2 + 8)(5x^2 - 8)$

Now, let's analyze option C. This expression is a product of two binomials, and it is not immediately clear whether it is equivalent to the given expression. To determine whether it is equivalent, we need to simplify the expression.

$(5x^2 + 8)(5x^2 - 8)$

We can start by multiplying the two binomials:

$25x^4 - 64$

This expression is equivalent to the given expression $25x^4 - 64$. Therefore, option C is equivalent to the given expression.

Option D: $(x^2 + 13)(x^2 - 13)$

Next, let's analyze option D. This expression is a product of two binomials, and it is not immediately clear whether it is equivalent to the given expression. To determine whether it is equivalent, we need to simplify the expression.

$(x^2 + 13)(x^2 - 13)$

We can start by multiplying the two binomials:

$x^4 - 169$

This expression is not equivalent to the given expression $25x^4 - 64$. The reason is that the constant term is -169, whereas the constant term in the given expression is -64. Therefore, option D is not equivalent to the given expression.

Option E: $(5x^2 - 8)^2$

Finally, let's analyze option E. This expression is a square of a binomial, and it is not immediately clear whether it is equivalent to the given expression. To determine whether it is equivalent, we need to simplify the expression.

$(5x^2 - 8)^2$

We can start by expanding the square:

$25x^4 - 80x^2 + 64$

This expression is not equivalent to the given expression $25x^4 - 64$. The reason is that the middle term is -80x^2, whereas there is no middle term in the given expression. Therefore, option E is not equivalent to the given expression.

Conclusion

In conclusion, only option C is equivalent to the given expression $25x^4 - 64$. The other options are not equivalent to the given expression. This is because option A and option B have the wrong exponent of $x$, option D has the wrong constant term, and option E has an extra middle term.

Key Takeaways

• Equivalent expressions have the same value for all possible values of the variable.
• To determine whether an expression is equivalent to a given expression, we need to simplify the expression and compare it to the given expression.
• In this article, we analyzed five options and determined which one is equivalent to the given expression $25x^4 - 64$.

Final Answer

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In the previous article, we explored which expressions are equivalent to the given expression $25x^4 - 64$. We analyzed five options and determined that only option C is equivalent to the given expression. In this article, we will answer some frequently asked questions about equivalent expressions.

Q: What is an equivalent expression?

A: An equivalent expression is an expression that has the same value for all possible values of the variable. In other words, equivalent expressions are expressions that are equal to each other.

Q: How do I determine whether an expression is equivalent to a given expression?

A: To determine whether an expression is equivalent to a given expression, you need to simplify the expression and compare it to the given expression. You can use various techniques such as combining like terms, factoring, and expanding to simplify the expression.

Q: What are some common techniques for simplifying expressions?

A: Some common techniques for simplifying expressions include:

• Combining like terms: This involves combining terms that have the same variable and exponent.
• Factoring: This involves expressing an expression as a product of simpler expressions.
• Expanding: This involves expressing an expression as a sum of simpler expressions.

Q: How do I know whether an expression is equivalent to a given expression?

A: To determine whether an expression is equivalent to a given expression, you need to check whether the expression has the same value for all possible values of the variable. You can do this by substituting different values of the variable into the expression and checking whether the result is the same as the given expression.

Q: What are some common mistakes to avoid when working with equivalent expressions?

A: Some common mistakes to avoid when working with equivalent expressions include:

• Not simplifying the expression enough: This can lead to incorrect conclusions about the equivalence of the expression.
• Not checking the expression for all possible values of the variable: This can lead to incorrect conclusions about the equivalence of the expression.
• Not using the correct techniques for simplifying the expression: This can lead to incorrect conclusions about the equivalence of the expression.

Q: How do I apply equivalent expressions in real-world problems?

A: Equivalent expressions can be applied in a variety of real-world problems, such as:

• Simplifying complex expressions: Equivalent expressions can be used to simplify complex expressions and make them easier to work with.
• Solving equations: Equivalent expressions can be used to solve equations by simplifying the expression and finding the solution.
• Graphing functions: Equivalent expressions can be used to graph functions by simplifying the expression and finding the graph.

Q: What are some common applications of equivalent expressions?

A: Some common applications of equivalent expressions include:

• Algebra: Equivalent expressions are used extensively in algebra to simplify complex expressions and solve equations.
• Calculus: Equivalent expressions are used in calculus to simplify complex expressions and find derivatives and integrals.
• Physics: Equivalent expressions are used in physics to simplify complex expressions and solve problems involving motion and energy.

Conclusion

In conclusion, equivalent expressions are an important concept in mathematics that can be used to simplify complex expressions and solve equations. By understanding how to determine whether an expression is equivalent to a given expression, you can apply equivalent expressions in a variety of real-world problems. Remember to avoid common mistakes and use the correct techniques for simplifying the expression.

Key Takeaways

• Equivalent expressions have the same value for all possible values of the variable.
• To determine whether an expression is equivalent to a given expression, you need to simplify the expression and compare it to the given expression.
• Equivalent expressions can be applied in a variety of real-world problems, such as simplifying complex expressions, solving equations, and graphing functions.

Final Answer

The final answer is that equivalent expressions are an important concept in mathematics that can be used to simplify complex expressions and solve equations. By understanding how to determine whether an expression is equivalent to a given expression, you can apply equivalent expressions in a variety of real-world problems.