Simplify The Expression: ${ \frac{4^{n-1} \times 8^{n+1} \times \frac{1}{2}}{32^n} }$

$$

Introduction

Simplifying algebraic expressions is a crucial skill in mathematics, and it requires a deep understanding of the underlying concepts. In this article, we will focus on simplifying a complex expression involving exponents and fractions. The given expression is $\frac{4^{n-1} \times 8^{n+1} \times \frac{1}{2}}{32^n}$, and our goal is to simplify it to its simplest form.

Understanding Exponents and Fractions

Before we dive into simplifying the expression, let's review some basic concepts related to exponents and fractions. Exponents are a shorthand way of representing repeated multiplication. For example, $a^b$ means $a$ multiplied by itself $b$ times. On the other hand, fractions are a way of representing a part of a whole. In this expression, we have a fraction with a numerator and a denominator.

Simplifying the Expression

To simplify the expression, we need to apply the rules of exponents and fractions. Let's start by simplifying the numerator. We can rewrite $8^{n+1}$ as $(2^3)^{n+1}$, which is equal to $2^{3(n+1)}$. Similarly, we can rewrite $4^{n-1}$ as $(2^2)^{n-1}$, which is equal to $2^{2(n-1)}$. Now, we can substitute these expressions back into the original expression.

Applying the Rules of Exponents

Now that we have simplified the numerator, let's apply the rules of exponents to simplify the expression further. We can start by combining the exponents in the numerator using the rule $a^b \times a^c = a^{b+c}$. This gives us $2^{2(n-1)} \times 2^{3(n+1)} \times \frac{1}{2}$. Next, we can simplify the fraction by multiplying the numerator and denominator by the reciprocal of the denominator.

Simplifying the Fraction

To simplify the fraction, we need to multiply the numerator and denominator by the reciprocal of the denominator. The reciprocal of $\frac{1}{2}$ is $2$, so we can multiply the numerator and denominator by $2$. This gives us $2^{2(n-1)} \times 2^{3(n+1)} \times 2$.

Applying the Product of Powers Rule

Now that we have simplified the fraction, let's apply the product of powers rule to simplify the expression further. The product of powers rule states that $a^b \times a^c = a^{b+c}$. We can apply this rule to the expression $2^{2(n-1)} \times 2^{3(n+1)} \times 2$.

Simplifying the Expression Further

Using the product of powers rule, we can simplify the expression as follows:

$2^{2(n-1)} \times 2^{3(n+1)} \times 2 = 2^{2(n-1) + 3(n+1) + 1}$

Evaluating the Exponent

Now that we have simplified the expression, let's evaluate the exponent. We can start by simplifying the expression inside the exponent.

$2(n-1) + 3(n+1) + 1 = 2n - 2 + 3n + 3 + 1$

Combining Like Terms

Next, we can combine like terms in the expression.

$2n - 2 + 3n + 3 + 1 = 5n + 2$

Simplifying the Expression

Now that we have evaluated the exponent, let's simplify the expression further. We can rewrite the expression as $2^{5n+2}$.

Simplifying the Denominator

Now that we have simplified the numerator, let's simplify the denominator. We can rewrite $32^n$ as $(2^5)^n$, which is equal to $2^{5n}$.

Simplifying the Expression

Now that we have simplified the numerator and denominator, let's simplify the expression further. We can rewrite the expression as $\frac{2^{5n+2}}{2^{5n}}$.

Applying the Quotient of Powers Rule

Now that we have simplified the expression, let's apply the quotient of powers rule to simplify it further. The quotient of powers rule states that $\frac{a^b}{a^c} = a^{b-c}$. We can apply this rule to the expression $\frac{2^{5n+2}}{2^{5n}}$.

Simplifying the Expression Further

Using the quotient of powers rule, we can simplify the expression as follows:

$\frac{2^{5n+2}}{2^{5n}} = 2^{(5n+2)-5n}$

Evaluating the Exponent

Now that we have simplified the expression, let's evaluate the exponent. We can start by simplifying the expression inside the exponent.

$(5n+2)-5n = 2$

Simplifying the Expression

Now that we have evaluated the exponent, let's simplify the expression further. We can rewrite the expression as $2^2$.

Conclusion

In this article, we simplified the expression $\frac{4^{n-1} \times 8^{n+1} \times \frac{1}{2}}{32^n}$ to its simplest form. We applied the rules of exponents and fractions to simplify the expression, and we used the product of powers rule and the quotient of powers rule to simplify it further. The final simplified expression is $2^2$.

Final Answer

The final answer is $\boxed{4}$.

$$

Introduction

In our previous article, we simplified the expression $\frac{4^{n-1} \times 8^{n+1} \times \frac{1}{2}}{32^n}$ to its simplest form. We applied the rules of exponents and fractions to simplify the expression, and we used the product of powers rule and the quotient of powers rule to simplify it further. In this article, we will answer some common questions related to the simplification of the expression.

Q&A

Q: What is the final simplified expression?

A: The final simplified expression is $2^2$.

Q: How did you simplify the expression?

A: We applied the rules of exponents and fractions to simplify the expression, and we used the product of powers rule and the quotient of powers rule to simplify it further.

Q: What is the product of powers rule?

A: The product of powers rule states that $a^b \times a^c = a^{b+c}$.

Q: What is the quotient of powers rule?

A: The quotient of powers rule states that $\frac{a^b}{a^c} = a^{b-c}$.

Q: How do you apply the product of powers rule?

A: To apply the product of powers rule, you need to multiply the exponents of the two expressions. For example, $2^3 \times 2^4 = 2^{3+4} = 2^7$.

Q: How do you apply the quotient of powers rule?

A: To apply the quotient of powers rule, you need to subtract the exponent of the denominator from the exponent of the numerator. For example, $\frac{2^5}{2^3} = 2^{5-3} = 2^2$.

Q: What is the difference between the product of powers rule and the quotient of powers rule?

A: The product of powers rule is used to multiply two expressions with the same base, while the quotient of powers rule is used to divide two expressions with the same base.

Q: Can you provide an example of how to simplify an expression using the product of powers rule?

A: Yes, here is an example:

$\frac{2^3 \times 2^4}{2^2} = \frac{2^{3+4}}{2^2} = \frac{2^7}{2^2} = 2^{7-2} = 2^5$

Q: Can you provide an example of how to simplify an expression using the quotient of powers rule?

A: Yes, here is an example:

$\frac{2^5}{2^3} = 2^{5-3} = 2^2$

Q: What is the importance of simplifying expressions?

A: Simplifying expressions is important because it helps to make the expression easier to understand and work with. It also helps to avoid errors and make calculations more efficient.

Conclusion

In this article, we answered some common questions related to the simplification of the expression $\frac{4^{n-1} \times 8^{n+1} \times \frac{1}{2}}{32^n}$. We provided examples of how to apply the product of powers rule and the quotient of powers rule, and we discussed the importance of simplifying expressions.

Final Answer

The final answer is $\boxed{4}$.