Simplify The Expression:${ \frac{4^{n-1} \cdot 24^{-2n} \cdot 4}{6^{-n} \cdot 3^{-n}} }$

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Introduction

In algebra, simplifying expressions is a crucial skill that helps us solve equations and inequalities more efficiently. It involves manipulating the given expression to make it easier to work with, often by combining like terms, canceling out common factors, or using properties of exponents. In this article, we will focus on simplifying the expression 4n−1⋅24−2n⋅46−n⋅3−n\frac{4^{n-1} \cdot 24^{-2n} \cdot 4}{6^{-n} \cdot 3^{-n}} using various algebraic techniques.

Understanding the Expression

Before we dive into simplifying the expression, let's break it down and understand its components. The expression consists of several terms, each with its own set of exponents and bases. We have:

  • 4n−14^{n-1}: This term has a base of 4 and an exponent of n−1n-1.
  • 24−2n24^{-2n}: This term has a base of 24 and an exponent of −2n-2n.
  • 44: This is a constant term with an exponent of 1.
  • 6−n6^{-n}: This term has a base of 6 and an exponent of −n-n.
  • 3−n3^{-n}: This term has a base of 3 and an exponent of −n-n.

Simplifying the Expression

To simplify the expression, we can start by combining like terms and canceling out common factors. We can rewrite the expression as follows:

4n−1⋅24−2n⋅46−n⋅3−n=4n⋅24−2n6−n⋅3−n\frac{4^{n-1} \cdot 24^{-2n} \cdot 4}{6^{-n} \cdot 3^{-n}} = \frac{4^{n} \cdot 24^{-2n}}{6^{-n} \cdot 3^{-n}}

Using Properties of Exponents

Now, let's use the properties of exponents to simplify the expression further. We can rewrite the expression as follows:

4n⋅24−2n6−n⋅3−n=4n⋅(3⋅8)−2n6−n⋅3−n\frac{4^{n} \cdot 24^{-2n}}{6^{-n} \cdot 3^{-n}} = \frac{4^{n} \cdot (3 \cdot 8)^{-2n}}{6^{-n} \cdot 3^{-n}}

Simplifying the Term Inside the Parentheses

We can simplify the term inside the parentheses by using the property of exponents that states (ab)n=anâ‹…bn(ab)^n = a^n \cdot b^n. We can rewrite the expression as follows:

4n⋅(3⋅8)−2n6−n⋅3−n=4n⋅3−2n⋅8−2n6−n⋅3−n\frac{4^{n} \cdot (3 \cdot 8)^{-2n}}{6^{-n} \cdot 3^{-n}} = \frac{4^{n} \cdot 3^{-2n} \cdot 8^{-2n}}{6^{-n} \cdot 3^{-n}}

Canceling Out Common Factors

Now, let's cancel out common factors between the numerator and denominator. We can rewrite the expression as follows:

4n⋅3−2n⋅8−2n6−n⋅3−n=4n⋅3−n⋅8−2n6−n\frac{4^{n} \cdot 3^{-2n} \cdot 8^{-2n}}{6^{-n} \cdot 3^{-n}} = \frac{4^{n} \cdot 3^{-n} \cdot 8^{-2n}}{6^{-n}}

Simplifying the Term with the Base of 8

We can simplify the term with the base of 8 by using the property of exponents that states (am)n=amâ‹…n(a^m)^n = a^{m \cdot n}. We can rewrite the expression as follows:

4n⋅3−n⋅8−2n6−n=4n⋅3−n⋅(23)−2n6−n\frac{4^{n} \cdot 3^{-n} \cdot 8^{-2n}}{6^{-n}} = \frac{4^{n} \cdot 3^{-n} \cdot (2^3)^{-2n}}{6^{-n}}

Simplifying the Term with the Base of 2

We can simplify the term with the base of 2 by using the property of exponents that states (am)n=amâ‹…n(a^m)^n = a^{m \cdot n}. We can rewrite the expression as follows:

4n⋅3−n⋅(23)−2n6−n=4n⋅3−n⋅2−6n6−n\frac{4^{n} \cdot 3^{-n} \cdot (2^3)^{-2n}}{6^{-n}} = \frac{4^{n} \cdot 3^{-n} \cdot 2^{-6n}}{6^{-n}}

Canceling Out Common Factors Again

Now, let's cancel out common factors between the numerator and denominator again. We can rewrite the expression as follows:

4n⋅3−n⋅2−6n6−n=4n⋅2−6n6−n\frac{4^{n} \cdot 3^{-n} \cdot 2^{-6n}}{6^{-n}} = \frac{4^{n} \cdot 2^{-6n}}{6^{-n}}

Simplifying the Term with the Base of 4

We can simplify the term with the base of 4 by using the property of exponents that states (am)n=amâ‹…n(a^m)^n = a^{m \cdot n}. We can rewrite the expression as follows:

4n⋅2−6n6−n=(22)n⋅2−6n6−n\frac{4^{n} \cdot 2^{-6n}}{6^{-n}} = \frac{(2^2)^{n} \cdot 2^{-6n}}{6^{-n}}

Simplifying the Term with the Base of 2 Again

We can simplify the term with the base of 2 again by using the property of exponents that states (am)n=amâ‹…n(a^m)^n = a^{m \cdot n}. We can rewrite the expression as follows:

(22)n⋅2−6n6−n=22n⋅2−6n6−n\frac{(2^2)^{n} \cdot 2^{-6n}}{6^{-n}} = \frac{2^{2n} \cdot 2^{-6n}}{6^{-n}}

Canceling Out Common Factors Once More

Now, let's cancel out common factors between the numerator and denominator once more. We can rewrite the expression as follows:

22n⋅2−6n6−n=2−4n6−n\frac{2^{2n} \cdot 2^{-6n}}{6^{-n}} = \frac{2^{-4n}}{6^{-n}}

Simplifying the Term with the Base of 2 Again

We can simplify the term with the base of 2 again by using the property of exponents that states (am)n=amâ‹…n(a^m)^n = a^{m \cdot n}. We can rewrite the expression as follows:

2−4n6−n=2−4n(2⋅3)−n\frac{2^{-4n}}{6^{-n}} = \frac{2^{-4n}}{(2 \cdot 3)^{-n}}

Canceling Out Common Factors One Last Time

Now, let's cancel out common factors between the numerator and denominator one last time. We can rewrite the expression as follows:

2−4n(2⋅3)−n=2−4n2−n⋅3−n\frac{2^{-4n}}{(2 \cdot 3)^{-n}} = \frac{2^{-4n}}{2^{-n} \cdot 3^{-n}}

Simplifying the Expression Finally

We can simplify the expression finally by canceling out common factors between the numerator and denominator. We can rewrite the expression as follows:

2−4n2−n⋅3−n=2−3n⋅3n\frac{2^{-4n}}{2^{-n} \cdot 3^{-n}} = 2^{-3n} \cdot 3^{n}

Conclusion

In this article, we simplified the expression 4n−1⋅24−2n⋅46−n⋅3−n\frac{4^{n-1} \cdot 24^{-2n} \cdot 4}{6^{-n} \cdot 3^{-n}} using various algebraic techniques. We combined like terms, canceled out common factors, and used properties of exponents to simplify the expression. The final simplified expression is 2−3n⋅3n2^{-3n} \cdot 3^{n}.

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Introduction

In our previous article, we simplified the expression 4n−1⋅24−2n⋅46−n⋅3−n\frac{4^{n-1} \cdot 24^{-2n} \cdot 4}{6^{-n} \cdot 3^{-n}} using various algebraic techniques. In this article, we will provide a Q&A guide to help you understand the steps involved in simplifying the expression.

Q&A: Simplifying the Expression

Q: What is the first step in simplifying the expression?

A: The first step in simplifying the expression is to combine like terms and cancel out common factors.

Q: How do I combine like terms in the expression?

A: To combine like terms, we need to identify the terms with the same base and exponent. In this case, we have 4n−14^{n-1} and 44, which can be combined as 4n4^{n}.

Q: What is the next step in simplifying the expression?

A: The next step is to use the properties of exponents to simplify the expression further. We can rewrite the expression as 4n⋅24−2n6−n⋅3−n\frac{4^{n} \cdot 24^{-2n}}{6^{-n} \cdot 3^{-n}}.

Q: How do I simplify the term inside the parentheses?

A: To simplify the term inside the parentheses, we can use the property of exponents that states (ab)n=an⋅bn(ab)^n = a^n \cdot b^n. In this case, we have (3⋅8)−2n(3 \cdot 8)^{-2n}, which can be rewritten as 3−2n⋅8−2n3^{-2n} \cdot 8^{-2n}.

Q: What is the next step in simplifying the expression?

A: The next step is to cancel out common factors between the numerator and denominator. We can rewrite the expression as 4n⋅3−2n⋅8−2n6−n⋅3−n\frac{4^{n} \cdot 3^{-2n} \cdot 8^{-2n}}{6^{-n} \cdot 3^{-n}}.

Q: How do I simplify the term with the base of 8?

A: To simplify the term with the base of 8, we can use the property of exponents that states (am)n=am⋅n(a^m)^n = a^{m \cdot n}. In this case, we have 8−2n8^{-2n}, which can be rewritten as (23)−2n(2^3)^{-2n}.

Q: What is the next step in simplifying the expression?

A: The next step is to simplify the term with the base of 2. We can rewrite the expression as 4n⋅3−n⋅2−6n6−n\frac{4^{n} \cdot 3^{-n} \cdot 2^{-6n}}{6^{-n}}.

Q: How do I cancel out common factors between the numerator and denominator?

A: To cancel out common factors, we need to identify the terms with the same base and exponent. In this case, we have 4n4^{n} and 6−n6^{-n}, which can be canceled out as 4n⋅6n4^{n} \cdot 6^{n}.

Q: What is the final simplified expression?

A: The final simplified expression is 2−3n⋅3n2^{-3n} \cdot 3^{n}.

Conclusion

In this article, we provided a Q&A guide to help you understand the steps involved in simplifying the expression 4n−1⋅24−2n⋅46−n⋅3−n\frac{4^{n-1} \cdot 24^{-2n} \cdot 4}{6^{-n} \cdot 3^{-n}}. We hope this guide has been helpful in simplifying the expression and understanding the properties of exponents.

Frequently Asked Questions

Q: What is the most important step in simplifying the expression?

A: The most important step in simplifying the expression is to combine like terms and cancel out common factors.

Q: How do I know when to use the properties of exponents?

A: You should use the properties of exponents when you have terms with the same base and exponent.

Q: What is the final simplified expression?

A: The final simplified expression is 2−3n⋅3n2^{-3n} \cdot 3^{n}.

Q: How do I check my work?

A: You should check your work by plugging in a value for nn and simplifying the expression.

Additional Resources

Conclusion

In conclusion, simplifying the expression 4n−1⋅24−2n⋅46−n⋅3−n\frac{4^{n-1} \cdot 24^{-2n} \cdot 4}{6^{-n} \cdot 3^{-n}} requires a combination of algebraic techniques, including combining like terms, canceling out common factors, and using properties of exponents. We hope this Q&A guide has been helpful in simplifying the expression and understanding the properties of exponents.