Evaluate The Expression $(4xy)^2$.

Mar 10, 2025 by ADMIN 37 views

Evaluate the Expression $(4xy)^2$

Introduction

In mathematics, expressions are a fundamental concept that helps us represent mathematical relationships and operations. Evaluating an expression involves simplifying it to a single value or a simpler form. In this article, we will focus on evaluating the expression $(4xy)^2$ . This involves applying the rules of exponents and simplifying the resulting expression.

Understanding Exponents

Before we dive into evaluating the expression, let's quickly review what exponents are. An exponent is a small number that is written to the upper right of a number or a variable. It represents the power to which the base is raised. For example, in the expression $x^2$ , the exponent 2 represents the power to which the base $x$ is raised.

Evaluating the Expression

To evaluate the expression $(4xy)^2$ , we need to apply the rule of exponents that states $(ab)^n = a^n \cdot b^n$ . In this case, the base is $4xy$ and the exponent is 2.

(4xy)^2 = (4xy) * (4xy)

Using the rule of exponents, we can rewrite the expression as:

(4xy)^2 = 4^2 * x^2 * y^2

Simplifying the Expression

Now that we have applied the rule of exponents, we can simplify the expression further. We know that $4^2 = 16$ , so we can substitute this value into the expression.

(4xy)^2 = 16 * x^2 * y^2

Final Answer

The final answer to the expression $(4xy)^2$ is $16x^2y^2$ . This is the simplified form of the expression after applying the rule of exponents and simplifying the resulting expression.

Conclusion

Evaluating expressions is an essential skill in mathematics that helps us simplify complex expressions and represent mathematical relationships in a concise form. In this article, we evaluated the expression $(4xy)^2$ using the rule of exponents and simplified the resulting expression to $16x^2y^2$ . This demonstrates the importance of understanding exponents and applying the rules of exponents to simplify complex expressions.

Additional Examples

Here are a few more examples of evaluating expressions using the rule of exponents:

$(3ab)^3 = 3^3 * a^3 * b^3 = 27a^3b^3$
$(2x)^4 = 2^4 * x^4 = 16x^4$
$(5y)^2 = 5^2 * y^2 = 25y^2$

These examples demonstrate how the rule of exponents can be applied to simplify complex expressions and represent mathematical relationships in a concise form.

Tips and Tricks

Here are a few tips and tricks to help you evaluate expressions using the rule of exponents:

Make sure to apply the rule of exponents correctly by following the order of operations (PEMDAS).
Use the rule of exponents to simplify complex expressions and represent mathematical relationships in a concise form.
Practice evaluating expressions using the rule of exponents to become more comfortable with the concept.

Frequently Asked Questions

Here are a few frequently asked questions about evaluating expressions using the rule of exponents:

Q: What is the rule of exponents? A: The rule of exponents states that $(ab)^n = a^n \cdot b^n$ .
Q: How do I apply the rule of exponents to simplify an expression? A: To apply the rule of exponents, simply rewrite the expression using the rule and simplify the resulting expression.
Q: What are some common mistakes to avoid when evaluating expressions using the rule of exponents? A: Some common mistakes to avoid include forgetting to apply the rule of exponents, not following the order of operations (PEMDAS), and not simplifying the resulting expression.

Conclusion

Introduction

Evaluating expressions is a fundamental concept in mathematics that helps us simplify complex expressions and represent mathematical relationships in a concise form. In our previous article, we evaluated the expression $(4xy)^2$ using the rule of exponents and simplified the resulting expression to $16x^2y^2$ . In this article, we will provide a Q&A guide to help you better understand evaluating expressions and apply the rule of exponents to simplify complex expressions.

Q&A Guide

Q: What is the rule of exponents?

A: The rule of exponents states that $(ab)^n = a^n \cdot b^n$ . This rule helps us simplify complex expressions by breaking them down into smaller parts.

Q: How do I apply the rule of exponents to simplify an expression?

A: To apply the rule of exponents, simply rewrite the expression using the rule and simplify the resulting expression. For example, if we have the expression $(3ab)^3$ , we can apply the rule of exponents by rewriting it as $3^3 \cdot a^3 \cdot b^3$ .

Q: What are some common mistakes to avoid when evaluating expressions using the rule of exponents?

A: Some common mistakes to avoid include forgetting to apply the rule of exponents, not following the order of operations (PEMDAS), and not simplifying the resulting expression.

Q: How do I handle negative exponents?

A: When we have a negative exponent, we can rewrite it as a positive exponent by taking the reciprocal of the base. For example, if we have the expression $x^{-2}$ , we can rewrite it as $\frac{1}{x^2}$ .

Q: Can I apply the rule of exponents to expressions with multiple variables?

A: Yes, you can apply the rule of exponents to expressions with multiple variables. For example, if we have the expression $(2xy)^3$ , we can apply the rule of exponents by rewriting it as $2^3 \cdot x^3 \cdot y^3$ .

Q: How do I handle expressions with fractional exponents?

A: When we have an expression with a fractional exponent, we can rewrite it as a product of two expressions. For example, if we have the expression $x^{\frac{1}{2}}$ , we can rewrite it as $\sqrt{x}$ .

Q: Can I apply the rule of exponents to expressions with radicals?

A: Yes, you can apply the rule of exponents to expressions with radicals. For example, if we have the expression $(\sqrt{x})^2$ , we can apply the rule of exponents by rewriting it as $x$ .

Q: How do I handle expressions with absolute values?

A: When we have an expression with an absolute value, we can rewrite it as a product of two expressions. For example, if we have the expression $|x|^2$ , we can rewrite it as $x^2$ .

Q: Can I apply the rule of exponents to expressions with trigonometric functions?

A: Yes, you can apply the rule of exponents to expressions with trigonometric functions. For example, if we have the expression $(\sin x)^2$ , we can apply the rule of exponents by rewriting it as $\sin^2 x$ .

Conclusion

Evaluating expressions is a fundamental concept in mathematics that helps us simplify complex expressions and represent mathematical relationships in a concise form. In this article, we provided a Q&A guide to help you better understand evaluating expressions and apply the rule of exponents to simplify complex expressions. By following the rule of exponents and avoiding common mistakes, you can simplify complex expressions and represent mathematical relationships in a concise form.

Additional Resources

Here are some additional resources to help you learn more about evaluating expressions and applying the rule of exponents:

Khan Academy: Evaluating Expressions
Mathway: Evaluating Expressions
Wolfram Alpha: Evaluating Expressions

Tips and Tricks

Here are a few tips and tricks to help you evaluate expressions using the rule of exponents:

Make sure to apply the rule of exponents correctly by following the order of operations (PEMDAS).
Use the rule of exponents to simplify complex expressions and represent mathematical relationships in a concise form.
Practice evaluating expressions using the rule of exponents to become more comfortable with the concept.

Frequently Asked Questions

Here are a few frequently asked questions about evaluating expressions using the rule of exponents:

Q: What is the rule of exponents? A: The rule of exponents states that $(ab)^n = a^n \cdot b^n$ .
Q: How do I apply the rule of exponents to simplify an expression? A: To apply the rule of exponents, simply rewrite the expression using the rule and simplify the resulting expression.
Q: What are some common mistakes to avoid when evaluating expressions using the rule of exponents? A: Some common mistakes to avoid include forgetting to apply the rule of exponents, not following the order of operations (PEMDAS), and not simplifying the resulting expression.