Which Expressions Are Equivalent To $25x^4 - 64$? Select Three Options.A. $25x^4 + 40x - 40x - 64$B. $ 25 X 4 + 13 X − 13 X − 64 25x^4 + 13x - 13x - 64 25 X 4 + 13 X − 13 X − 64 [/tex]C. $\left(5x^2 + 8\right)\left(5x^2 - 8\right)$D. $\left(x^2 +

Mar 10, 2025 by ADMIN 254 views

Which Expressions are Equivalent to $25x^4 - 64$ ?

Understanding the Problem

When dealing with algebraic expressions, it's essential to identify equivalent expressions that can be simplified or factored to reveal their underlying structure. In this case, we're given the expression $25x^4 - 64$ and asked to select three equivalent options from the provided choices.

Option A: $25x^4 + 40x - 40x - 64$

At first glance, option A may seem like a plausible choice, but let's examine it more closely. The expression $25x^4 + 40x - 40x - 64$ can be simplified by combining like terms. However, the presence of the $40x$ and $-40x$ terms suggests that this expression is not equivalent to the original expression $25x^4 - 64$ . When we combine the like terms, we get $25x^4 - 64$ , which is indeed the original expression. However, this is not the only way to simplify the expression, and we need to consider other options as well.

Option B: $25x^4 + 13x - 13x - 64$

Similar to option A, option B also contains like terms that can be combined. However, the presence of the $13x$ and $-13x$ terms suggests that this expression is not equivalent to the original expression $25x^4 - 64$ . When we combine the like terms, we get $25x^4 - 64$ , which is indeed the original expression. However, this is not the only way to simplify the expression, and we need to consider other options as well.

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

Now, let's examine option C, which involves the product of two binomials. When we multiply the two binomials, we get:

$\left(5x^2 + 8\right)\left(5x^2 - 8\right) = \left(5x^2\right)^2 - \left(8\right)^2$

Using the difference of squares formula, we can simplify this expression as follows:

$\left(5x^2\right)^2 - \left(8\right)^2 = 25x^4 - 64$

This expression is indeed equivalent to the original expression $25x^4 - 64$ . Therefore, option C is a valid choice.

Option D: $\left(x^2 + 8\right)\left(x^2 - 8\right)$

Now, let's examine option D, which also involves the product of two binomials. When we multiply the two binomials, we get:

$\left(x^2 + 8\right)\left(x^2 - 8\right) = \left(x^2\right)^2 - \left(8\right)^2$

Using the difference of squares formula, we can simplify this expression as follows:

$\left(x^2\right)^2 - \left(8\right)^2 = x^4 - 64$

This expression is not equivalent to the original expression $25x^4 - 64$ . Therefore, option D is not a valid choice.

Conclusion

In conclusion, we have examined four options and determined that only option C is equivalent to the original expression $25x^4 - 64$ . The other options, while similar in structure, do not yield the same result when simplified. Therefore, the correct answer is:

C. $\left(5x^2 + 8\right)\left(5x^2 - 8\right)$

Understanding the Concept of Equivalent Expressions

Equivalent expressions are algebraic expressions that can be simplified or factored to reveal their underlying structure. In this case, we used the difference of squares formula to simplify the expression $\left(5x^2 + 8\right)\left(5x^2 - 8\right)$ and obtained the original expression $25x^4 - 64$ . This demonstrates the importance of understanding the concept of equivalent expressions in algebra.

Real-World Applications

The concept of equivalent expressions has numerous real-world applications in fields such as engineering, physics, and computer science. For example, in engineering, equivalent expressions can be used to simplify complex mathematical models and optimize system performance. In physics, equivalent expressions can be used to describe the behavior of physical systems and make predictions about their behavior. In computer science, equivalent expressions can be used to optimize algorithms and improve system performance.

Tips and Tricks

When working with equivalent expressions, it's essential to remember the following tips and tricks:

Use the difference of squares formula to simplify expressions of the form $\left(a + b\right)\left(a - b\right)$ .
Use the sum of squares formula to simplify expressions of the form $\left(a + b\right)^2$ .
Use the product of two binomials to simplify expressions of the form $\left(a + b\right)\left(c + d\right)$ .
Use the distributive property to simplify expressions of the form $a\left(b + c\right)$ .

By following these tips and tricks, you can simplify complex algebraic expressions and reveal their underlying structure.

Common Mistakes

When working with equivalent expressions, it's essential to avoid common mistakes such as:

Failing to simplify expressions using the difference of squares formula.
Failing to simplify expressions using the sum of squares formula.
Failing to simplify expressions using the product of two binomials.
Failing to simplify expressions using the distributive property.

By avoiding these common mistakes, you can ensure that your algebraic expressions are simplified correctly and reveal their underlying structure.

Conclusion

C. $\left(5x^2 + 8\right)\left(5x^2 - 8\right)$

By understanding the concept of equivalent expressions and using the difference of squares formula, we can simplify complex algebraic expressions and reveal their underlying structure.
Q&A: Equivalent Expressions

Q: What is an equivalent expression?

A: An equivalent expression is an algebraic expression that can be simplified or factored to reveal its underlying structure. Equivalent expressions have the same value, but may be expressed in different ways.

Q: How do I determine if two expressions are equivalent?

A: To determine if two expressions are equivalent, you can simplify both expressions using algebraic properties and formulas. If the simplified expressions are the same, then the original expressions are equivalent.

Q: What are some common algebraic properties and formulas used to simplify expressions?

A: Some common algebraic properties and formulas used to simplify expressions include:

The difference of squares formula: $\left(a + b\right)\left(a - b\right) = a^2 - b^2$
The sum of squares formula: $\left(a + b\right)^2 = a^2 + 2ab + b^2$
The product of two binomials: $\left(a + b\right)\left(c + d\right) = ac + ad + bc + bd$
The distributive property: $a\left(b + c\right) = ab + ac$

Q: How do I use the difference of squares formula to simplify expressions?

A: To use the difference of squares formula, you can identify expressions of the form $\left(a + b\right)\left(a - b\right)$ and simplify them using the formula. For example, if you have the expression $\left(5x^2 + 8\right)\left(5x^2 - 8\right)$ , you can simplify it using the difference of squares formula as follows:

$\left(5x^2 + 8\right)\left(5x^2 - 8\right) = \left(5x^2\right)^2 - \left(8\right)^2 = 25x^4 - 64$

Q: How do I use the sum of squares formula to simplify expressions?

A: To use the sum of squares formula, you can identify expressions of the form $\left(a + b\right)^2$ and simplify them using the formula. For example, if you have the expression $\left(3x + 4\right)^2$ , you can simplify it using the sum of squares formula as follows:

$\left(3x + 4\right)^2 = \left(3x\right)^2 + 2\left(3x\right)\left(4\right) + \left(4\right)^2 = 9x^2 + 24x + 16$

Q: How do I use the product of two binomials to simplify expressions?

A: To use the product of two binomials, you can identify expressions of the form $\left(a + b\right)\left(c + d\right)$ and simplify them using the formula. For example, if you have the expression $\left(2x + 3\right)\left(x + 4\right)$ , you can simplify it using the product of two binomials as follows:

$\left(2x + 3\right)\left(x + 4\right) = 2x^2 + 8x + 3x + 12 = 2x^2 + 11x + 12$

Q: How do I use the distributive property to simplify expressions?

A: To use the distributive property, you can identify expressions of the form $a\left(b + c\right)$ and simplify them using the formula. For example, if you have the expression $4\left(2x + 3\right)$ , you can simplify it using the distributive property as follows:

$4\left(2x + 3\right) = 4\left(2x\right) + 4\left(3\right) = 8x + 12$

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

A: Some common mistakes to avoid when simplifying expressions include:

Failing to simplify expressions using the difference of squares formula.
Failing to simplify expressions using the sum of squares formula.
Failing to simplify expressions using the product of two binomials.
Failing to simplify expressions using the distributive property.
Not combining like terms correctly.

Q: How do I know if an expression is equivalent to another expression?

A: To determine if an expression is equivalent to another expression, you can simplify both expressions using algebraic properties and formulas. If the simplified expressions are the same, then the original expressions are equivalent.

Q: Can you give an example of equivalent expressions?

A: Yes, here is an example of equivalent expressions:

$\left(5x^2 + 8\right)\left(5x^2 - 8\right) = 25x^4 - 64$

This expression is equivalent to the expression $25x^4 - 64$ because they have the same value.

Q: Can you give another example of equivalent expressions?

A: Yes, here is another example of equivalent expressions:

$\left(3x + 4\right)^2 = 9x^2 + 24x + 16$

This expression is equivalent to the expression $9x^2 + 24x + 16$ because they have the same value.

Conclusion

In conclusion, equivalent expressions are algebraic expressions that can be simplified or factored to reveal their underlying structure. By using algebraic properties and formulas, you can simplify expressions and determine if they are equivalent to other expressions. Remember to avoid common mistakes and combine like terms correctly to ensure that your expressions are simplified correctly.