Rewrite The Expression $\left(\frac{3}{4^5}\right)^2$ As A Quotient Of Two Powers.$\left(\frac{3}{4^5}\right)^2 = ?$

Introduction

In this article, we will explore the concept of rewriting an expression as a quotient of two powers. This involves applying the rules of exponents to simplify the given expression. We will focus on the expression $\left(\frac{3}{4^5}\right)^2$ and rewrite it as a quotient of two powers.

Understanding the Expression

The given expression is $\left(\frac{3}{4^5}\right)^2$. To rewrite this expression as a quotient of two powers, we need to apply the rules of exponents. The expression can be broken down into two parts: the numerator and the denominator.

• The numerator is 3, which is a constant.
• The denominator is $4^5$, which is a power of 4.

Applying the Rules of Exponents

To rewrite the expression as a quotient of two powers, we need to apply the following rules of exponents:

• Power of a Power Rule: $(a^m)^n = a^{mn}$
• Quotient of Powers Rule: $\frac{a^m}{a^n} = a^{m-n}$

We can start by applying the Power of a Power Rule to the denominator:

$$4^5 = (2^2)^5 = 2^{2 \cdot 5} = 2^{10}$$

Now, we can rewrite the original expression as:

$$\left(\frac{3}{4^5}\right)^2 = \left(\frac{3}{2^{10}}\right)^2$$

Simplifying the Expression

Next, we can apply the Power of a Power Rule to the entire expression:

$$\left(\frac{3}{2^{10}}\right)^2 = \frac{3^2}{(2^{10})^2} = \frac{9}{2^{20}}$$

Rewriting as a Quotient of Two Powers

Finally, we can rewrite the expression as a quotient of two powers:

$$\frac{9}{2^{20}} = \frac{3^2}{2^{20}} = \frac{3^2}{2^{2 \cdot 10}} = \frac{3^2}{2^{2 \cdot 10}}$$

Conclusion

In this article, we have rewritten the expression $\left(\frac{3}{4^5}\right)^2$ as a quotient of two powers. We applied the rules of exponents, including the Power of a Power Rule and the Quotient of Powers Rule, to simplify the expression. The final result is $\frac{3^2}{2^{20}}$.

Key Takeaways

• The Power of a Power Rule states that $(a^m)^n = a^{mn}$.
• The Quotient of Powers Rule states that $\frac{a^m}{a^n} = a^{m-n}$.
• To rewrite an expression as a quotient of two powers, we need to apply the rules of exponents.

Practice Problems

1. Rewrite the expression $\left(\frac{2}{3^4}\right)^3$ as a quotient of two powers.
2. Rewrite the expression $\left(\frac{5}{2^3}\right)^2$ as a quotient of two powers.

Solutions

1. $\left(\frac{2}{3^4}\right)^3 = \frac{2^3}{(3^4)^3} = \frac{8}{3^{12}}$
2. $\left(\frac{5}{2^3}\right)^2 = \frac{5^2}{(2^3)^2} = \frac{25}{2^6}$
   Rewrite the Expression as a Quotient of Two Powers: Q&A =====================================================

Introduction

In our previous article, we explored the concept of rewriting an expression as a quotient of two powers. We applied the rules of exponents to simplify the given expression and arrived at the final result. In this article, we will provide a Q&A section to help you better understand the concept and apply it to different problems.

Q&A

Q: What is the Power of a Power Rule?

A: The Power of a Power Rule states that $(a^m)^n = a^{mn}$. This rule allows us to simplify expressions by combining powers.

Q: How do I apply the Power of a Power Rule?

A: To apply the Power of a Power Rule, simply multiply the exponents of the base and the power. For example, $(2^3)^4 = 2^{3 \cdot 4} = 2^{12}$.

Q: What is the Quotient of Powers Rule?

A: The Quotient of Powers Rule states that $\frac{a^m}{a^n} = a^{m-n}$. This rule allows us to simplify expressions by subtracting the exponents.

Q: How do I apply the Quotient of Powers Rule?

A: To apply the Quotient of Powers Rule, simply subtract the exponents of the base. For example, $\frac{2^5}{2^3} = 2^{5-3} = 2^2$.

Q: How do I rewrite an expression as a quotient of two powers?

A: To rewrite an expression as a quotient of two powers, follow these steps:

1. Apply the Power of a Power Rule to the denominator.
2. Apply the Quotient of Powers Rule to simplify the expression.
3. Rewrite the expression as a quotient of two powers.

Q: What are some common mistakes to avoid when rewriting expressions as quotients of two powers?

A: Some common mistakes to avoid include:

• Forgetting to apply the Power of a Power Rule to the denominator.
• Forgetting to apply the Quotient of Powers Rule to simplify the expression.
• Not rewriting the expression as a quotient of two powers.

Q: How do I practice rewriting expressions as quotients of two powers?

A: To practice rewriting expressions as quotients of two powers, try the following:

• Start with simple expressions and gradually move on to more complex ones.
• Use online resources or practice problems to help you practice.
• Review the rules of exponents and practice applying them.

Common Mistakes

Common Mistakes to Avoid

• Forgetting to apply the Power of a Power Rule to the denominator.
• Forgetting to apply the Quotient of Powers Rule to simplify the expression.
• Not rewriting the expression as a quotient of two powers.

How to Avoid Common Mistakes

• Read the problem carefully and understand what is being asked.
• Apply the rules of exponents in the correct order.
• Rewrite the expression as a quotient of two powers.

Conclusion

In this article, we have provided a Q&A section to help you better understand the concept of rewriting an expression as a quotient of two powers. We have covered common mistakes to avoid and provided tips on how to practice rewriting expressions as quotients of two powers. By following these tips and practicing regularly, you will become more confident in your ability to rewrite expressions as quotients of two powers.

Practice Problems

1. Rewrite the expression $\left(\frac{3}{2^4}\right)^3$ as a quotient of two powers.
2. Rewrite the expression $\left(\frac{4}{5^2}\right)^2$ as a quotient of two powers.

Solutions

1. $\left(\frac{3}{2^4}\right)^3 = \frac{3^3}{(2^4)^3} = \frac{27}{2^{12}}$
2. $\left(\frac{4}{5^2}\right)^2 = \frac{4^2}{(5^2)^2} = \frac{16}{5^4}$