Select The Correct Answer.Use The Properties Of Exponents To Rewrite The Expression: ( 2 A 4 ) 2 A 0 A 5 \frac{(2a^4)^2}{a^0 A^5} A 0 A 5 ( 2 A 4 ) 2 ​ What Is The Value Of The Rewritten Expression When A = − 5 A = -5 A = − 5 ?A. -250 B. -20 C. -500 D. -60

Introduction

Exponents are a fundamental concept in mathematics, and understanding how to simplify exponential expressions is crucial for solving various mathematical problems. In this article, we will focus on using the properties of exponents to rewrite the expression $\frac{(2a^4)^2}{a^0 a^5}$ and then evaluate its value when $a = -5$.

Properties of Exponents

Before we dive into simplifying the given expression, let's review the properties of exponents that we will use:

• Product of Powers Property: $a^m \cdot a^n = a^{m+n}$
• Power of a Power Property: $(a^m)^n = a^{m \cdot n}$
• Quotient of Powers Property: $\frac{a^m}{a^n} = a^{m-n}$

Simplifying the Expression

Using the properties of exponents, we can simplify the given expression as follows:

$$\frac{(2a^4)^2}{a^0 a^5} = \frac{2^2 (a^4)^2}{a^0 a^5} = \frac{4a^8}{a^0 a^5}$$

Now, let's simplify the expression further by applying the quotient of powers property:

$$\frac{4a^8}{a^0 a^5} = 4a^{8-0-5} = 4a^3$$

Evaluating the Expression

Now that we have simplified the expression to $4a^3$, we can evaluate its value when $a = -5$.

$$4a^3 = 4(-5)^3 = 4(-125) = -500$$

Conclusion

In this article, we used the properties of exponents to rewrite the expression $\frac{(2a^4)^2}{a^0 a^5}$ and then evaluated its value when $a = -5$. We simplified the expression using the product of powers property, power of a power property, and quotient of powers property. Finally, we evaluated the expression to find its value when $a = -5$, which is $-500$.

Answer

The correct answer is C. -500.

Discussion

This problem requires a good understanding of the properties of exponents and how to apply them to simplify complex expressions. It also requires the ability to evaluate expressions with variables and apply mathematical operations to find the final value.

Additional Examples

Here are a few additional examples of simplifying exponential expressions using the properties of exponents:

• $\frac{(3a^2)^3}{a^4} = \frac{3^3 (a^2)^3}{a^4} = \frac{27a^6}{a^4} = 27a^{6-4} = 27a^2$
• $\frac{2^5 a^3}{a^2} = \frac{2^5 a^3}{a^2} = 2^5 a^{3-2} = 32a$

Q: What are the properties of exponents?

A: The properties of exponents are a set of rules that help us simplify complex exponential expressions. The three main properties of exponents are:

• Product of Powers Property: $a^m \cdot a^n = a^{m+n}$
• Power of a Power Property: $(a^m)^n = a^{m \cdot n}$
• Quotient of Powers Property: $\frac{a^m}{a^n} = a^{m-n}$

Q: How do I simplify an exponential expression using the product of powers property?

A: To simplify an exponential expression using the product of powers property, we multiply the exponents of the two expressions. For example:

$$a^m \cdot a^n = a^{m+n}$$

Q: How do I simplify an exponential expression using the power of a power property?

A: To simplify an exponential expression using the power of a power property, we multiply the exponents of the two expressions. For example:

$$(a^m)^n = a^{m \cdot n}$$

Q: How do I simplify an exponential expression using the quotient of powers property?

A: To simplify an exponential expression using the quotient of powers property, we subtract the exponents of the two expressions. For example:

$$\frac{a^m}{a^n} = a^{m-n}$$

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

A: A positive exponent indicates that the base is raised to a power, while a negative exponent indicates that the base is raised to a power and then taken as a reciprocal. For example:

$$a^m = a \cdot a \cdot a \cdot ... \cdot a \text{ (m times)}$$

$$a^{-m} = \frac{1}{a \cdot a \cdot a \cdot ... \cdot a \text{ (m times)}}$$

Q: How do I evaluate an exponential expression with a variable?

A: To evaluate an exponential expression with a variable, we substitute the value of the variable into the expression and then simplify. For example:

$$4a^3 = 4(-5)^3 = 4(-125) = -500$$

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

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

• Not applying the properties of exponents correctly
• Not simplifying the expression fully
• Not evaluating the expression with variables correctly

Q: How do I practice simplifying exponential expressions?

A: To practice simplifying exponential expressions, try the following:

• Start with simple expressions and gradually move on to more complex ones
• Use online resources or worksheets to practice simplifying exponential expressions
• Work with a partner or tutor to get feedback on your work

Conclusion

Simplifying exponential expressions is an important skill in mathematics, and understanding the properties of exponents is crucial for simplifying complex expressions. By following the steps outlined in this article and practicing regularly, you can become proficient in simplifying exponential expressions and evaluating their values when variables are given.