Simplify The Expression Below.$(2x)^4$A. $2x^4$ B. $ 6 X 4 6x^4 6 X 4 [/tex] C. $8x^4$ D. $16x^4$

Introduction

Exponential expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will focus on simplifying the expression $(2x)^4$, which is a common problem in algebra and mathematics. We will break down the solution into step-by-step instructions, using clear and concise language to ensure that readers understand the process.

Understanding Exponents

Before we dive into the solution, let's review the basics of exponents. An exponent is a small number that is written above and to the right of a base number. It indicates how many times the base number should be multiplied by itself. For example, in the expression $x^4$, the exponent 4 indicates that the base number $x$ should be multiplied by itself 4 times: $x \times x \times x \times x$.

Simplifying the Expression

Now that we have a basic understanding of exponents, let's simplify the expression $(2x)^4$. To do this, we need to apply the power rule of exponents, which states that $(ab)^n = a^n \times b^n$. In this case, the base number is $2x$, and the exponent is 4.

Using the power rule, we can rewrite the expression as:

$$(2x)^4 = 2^4 \times x^4$$

Evaluating the Exponent

Now that we have applied the power rule, let's evaluate the exponent $2^4$. This means that we need to multiply 2 by itself 4 times:

$$2^4 = 2 \times 2 \times 2 \times 2 = 16$$

Simplifying the Expression Further

Now that we have evaluated the exponent, let's simplify the expression further. We can rewrite the expression as:

$$2^4 \times x^4 = 16 \times x^4$$

Conclusion

In conclusion, simplifying the expression $(2x)^4$ involves applying the power rule of exponents and evaluating the resulting exponent. By following these step-by-step instructions, we can simplify the expression to $16x^4$. This is the correct answer, which can be verified by multiplying the base number $2x$ by itself 4 times.

Common Mistakes to Avoid

When simplifying exponential expressions, there are several common mistakes to avoid. These include:

• Forgetting to apply the power rule: This is a common mistake that can lead to incorrect answers. Make sure to apply the power rule whenever you see an exponent.
• Evaluating the exponent incorrectly: This can also lead to incorrect answers. Make sure to evaluate the exponent correctly by multiplying the base number by itself the correct number of times.
• Not simplifying the expression further: This can also lead to incorrect answers. Make sure to simplify the expression further by combining like terms.

Practice Problems

To practice simplifying exponential expressions, try the following problems:

• Simplify the expression $(3x)^5$.
• Simplify the expression $(4y)^3$.
• Simplify the expression $(2z)^2$.

Answer Key

Here are the answers to the practice problems:

• $(3x)^5 = 243x^5$
• $(4y)^3 = 64y^3$
• $(2z)^2 = 4z^2$

Conclusion

Introduction

In our previous article, we explored the concept of simplifying exponential expressions, focusing on the expression $(2x)^4$. We broke down the solution into step-by-step instructions, using clear and concise language to ensure that readers understand the process. In this article, we will continue to build on this concept by providing a Q&A guide to help readers better understand and apply the principles of simplifying exponential expressions.

Q&A Guide

Q: What is the power rule of exponents?

A: The power rule of exponents states that $(ab)^n = a^n \times b^n$. This means that when you have an expression with a base and an exponent, you can rewrite it as the base raised to the power of the exponent, multiplied by the base raised to the power of the exponent.

Q: How do I apply the power rule of exponents?

A: To apply the power rule of exponents, simply rewrite the expression as the base raised to the power of the exponent, multiplied by the base raised to the power of the exponent. For example, if you have the expression $(2x)^4$, you can rewrite it as $2^4 \times x^4$.

Q: What is the difference between an exponent and a coefficient?

A: An exponent is a small number that is written above and to the right of a base number, indicating how many times the base number should be multiplied by itself. A coefficient, on the other hand, is a number that is multiplied by a variable or expression. For example, in the expression $3x^4$, the 3 is a coefficient and the 4 is an exponent.

Q: How do I simplify an expression with multiple exponents?

A: To simplify an expression with multiple exponents, you can use the power rule of exponents to rewrite the expression as a product of powers. For example, if you have the expression $(2x)^4 \times (3y)^2$, you can rewrite it as $2^4 \times x^4 \times 3^2 \times y^2$.

Q: What is the difference between a positive exponent and a negative exponent?

A: A positive exponent indicates that the base number should be multiplied by itself a certain number of times. A negative exponent, on the other hand, indicates that the reciprocal of the base number should be multiplied by itself a certain number of times. For example, in the expression $x^{-2}$, the -2 indicates that the reciprocal of x should be multiplied by itself 2 times.

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

A: To simplify an expression with a negative exponent, you can rewrite it as the reciprocal of the base number raised to the power of the absolute value of the exponent. For example, if you have the expression $x^{-2}$, you can rewrite it as $\frac{1}{x^2}$.

Q: What is the difference between an exponential expression and a polynomial expression?

A: An exponential expression is an expression that contains a base and an exponent, such as $2^4$. A polynomial expression, on the other hand, is an expression that contains variables and coefficients, such as $3x^4 + 2x^2 + 1$.

Q: How do I simplify an exponential expression that is part of a larger polynomial expression?

A: To simplify an exponential expression that is part of a larger polynomial expression, you can use the power rule of exponents to rewrite the expression as a product of powers. For example, if you have the expression $3x^4 + 2x^2 + 2^4$, you can rewrite it as $3x^4 + 2x^2 + 16$.

Conclusion

In conclusion, simplifying exponential expressions is a crucial skill for students and professionals alike. By following the step-by-step instructions outlined in this article, we can simplify expressions like $(2x)^4$ to $16x^4$. Remember to apply the power rule, evaluate the exponent correctly, and simplify the expression further to avoid common mistakes. With practice, you will become proficient in simplifying exponential expressions and be able to tackle more complex problems with confidence.