Simplify The Expression:$4cd\left(2c^{60}d^4\right)^0$

Feb 26, 2025 by ADMIN 55 views

$Simplify the Expression: $4cd\left(2c^{60}d^4\right)^0$$

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us to solve problems more efficiently. When dealing with exponents, it's essential to understand the rules of exponentiation to simplify expressions. In this article, we will focus on simplifying the expression $4cd\left(2c^{60}d^4\right)^0$ . We will break down the expression, apply the rules of exponentiation, and simplify it step by step.

Understanding Exponents

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

Simplifying the Expression

Now that we have a basic understanding of exponents, let's simplify the expression $4cd\left(2c^{60}d^4\right)^0$ . To simplify this expression, we need to apply the rules of exponentiation.

Rule 1: Any number raised to the power of 0 is equal to 1

According to the rules of exponentiation, any number raised to the power of 0 is equal to 1. This means that $\left(2c^{60}d^4\right)^0 = 1$ .

Rule 2: When simplifying an expression with exponents, we can multiply the coefficients and add the exponents

Now that we have simplified the expression inside the parentheses, we can multiply the coefficients and add the exponents. The coefficient of the expression inside the parentheses is 1, so we can multiply it by the coefficient of the expression outside the parentheses, which is $4cd$ . This gives us $4cd \cdot 1 = 4cd$ .

Rule 3: When simplifying an expression with exponents, we can remove the parentheses

Now that we have multiplied the coefficients and added the exponents, we can remove the parentheses. This gives us $4cd$ .

Conclusion

In conclusion, we have simplified the expression $4cd\left(2c^{60}d^4\right)^0$ by applying the rules of exponentiation. We first simplified the expression inside the parentheses by raising it to the power of 0, which equals 1. Then, we multiplied the coefficients and added the exponents, which gave us $4cd$ . Finally, we removed the parentheses, which gave us the final simplified expression $4cd$ .

Final Answer

The final answer to the expression $4cd\left(2c^{60}d^4\right)^0$ is $\boxed{4cd}$ .

Frequently Asked Questions

Q: What is the rule for simplifying an expression with exponents?

A: The rule for simplifying an expression with exponents is to multiply the coefficients and add the exponents.

Q: What is the rule for any number raised to the power of 0?

A: The rule for any number raised to the power of 0 is that it equals 1.

Q: How do we simplify an expression with exponents?

A: To simplify an expression with exponents, we can multiply the coefficients and add the exponents, and then remove the parentheses.

Step-by-Step Solution

Step 1: Simplify the expression inside the parentheses

The expression inside the parentheses is $\left(2c^{60}d^4\right)^0$ . According to the rules of exponentiation, any number raised to the power of 0 is equal to 1. Therefore, $\left(2c^{60}d^4\right)^0 = 1$ .

Step 2: Multiply the coefficients and add the exponents

The coefficient of the expression inside the parentheses is 1, so we can multiply it by the coefficient of the expression outside the parentheses, which is $4cd$ . This gives us $4cd \cdot 1 = 4cd$ .

Step 3: Remove the parentheses

Now that we have multiplied the coefficients and added the exponents, we can remove the parentheses. This gives us $4cd$ .

Common Mistakes

Mistake 1: Not simplifying the expression inside the parentheses

When simplifying an expression with exponents, it's essential to simplify the expression inside the parentheses first. If we don't simplify the expression inside the parentheses, we may end up with an incorrect answer.

Mistake 2: Not multiplying the coefficients and adding the exponents

When simplifying an expression with exponents, we need to multiply the coefficients and add the exponents. If we don't do this, we may end up with an incorrect answer.

Mistake 3: Not removing the parentheses

When simplifying an expression with exponents, we need to remove the parentheses after multiplying the coefficients and adding the exponents. If we don't do this, we may end up with an incorrect answer.

Real-World Applications

Simplifying expressions with exponents has many real-world applications. For example, in physics, we use exponents to describe the behavior of particles and waves. In engineering, we use exponents to describe the behavior of electrical circuits and mechanical systems. In finance, we use exponents to describe the behavior of investments and returns.

Conclusion

In conclusion, simplifying expressions with exponents is a crucial skill that has many real-world applications. By understanding the rules of exponentiation and applying them correctly, we can simplify expressions and solve problems more efficiently. Whether you're a student, a professional, or just someone who loves math, simplifying expressions with exponents is an essential skill that you should master.

Introduction

In our previous article, we simplified the expression $4cd\left(2c^{60}d^4\right)^0$ by applying the rules of exponentiation. We broke down the expression, applied the rules of exponentiation, and simplified it step by step. In this article, we will answer some frequently asked questions about simplifying expressions with exponents.

Q&A

Q: What is the rule for simplifying an expression with exponents?

A: The rule for simplifying an expression with exponents is to multiply the coefficients and add the exponents.

Q: What is the rule for any number raised to the power of 0?

A: The rule for any number raised to the power of 0 is that it equals 1.

Q: How do we simplify an expression with exponents?

A: To simplify an expression with exponents, we can multiply the coefficients and add the exponents, and then remove the parentheses.

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

A: A coefficient is a number that is multiplied by a variable, while an exponent is a small number that is written above and to the right of a larger number, indicating how many times the larger number should be multiplied by itself.

Q: Can we simplify an expression with exponents if it has multiple variables?

A: Yes, we can simplify an expression with exponents if it has multiple variables. We just need to apply the rules of exponentiation and simplify the expression step by step.

Q: How do we 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, $a^{-b} = \frac{1}{a^b}$ .

Q: Can we simplify an expression with exponents if it has a fraction as a coefficient?

A: Yes, we can simplify an expression with exponents if it has a fraction as a coefficient. We just need to apply the rules of exponentiation and simplify the expression step by step.

Q: How do we handle exponents with different bases?

A: When we have exponents with different bases, we can simplify the expression by applying the rules of exponentiation. For example, $(a^b)^c = a^{bc}$ .

Q: Can we simplify an expression with exponents if it has a variable as a coefficient?

A: Yes, we can simplify an expression with exponents if it has a variable as a coefficient. We just need to apply the rules of exponentiation and simplify the expression step by step.

Real-World Applications

Conclusion

Final Answer

The final answer to the expression $4cd\left(2c^{60}d^4\right)^0$ is $\boxed{4cd}$ .

Step-by-Step Solution

Step 1: Simplify the expression inside the parentheses

Step 2: Multiply the coefficients and add the exponents

The coefficient of the expression inside the parentheses is 1, so we can multiply it by the coefficient of the expression outside the parentheses, which is $4cd$ . This gives us $4cd \cdot 1 = 4cd$ .

Step 3: Remove the parentheses

Now that we have multiplied the coefficients and added the exponents, we can remove the parentheses. This gives us $4cd$ .

Common Mistakes

Mistake 1: Not simplifying the expression inside the parentheses

Mistake 2: Not multiplying the coefficients and adding the exponents

When simplifying an expression with exponents, we need to multiply the coefficients and add the exponents. If we don't do this, we may end up with an incorrect answer.