Select The Correct Answer.Which Of The Following Is Equal To The Expression Below?$\[ (160 \cdot 243)^{\frac{1}{5}} \\]A. 96 B. 80 C. \[$5 \sqrt[5]{5}\$\] D. \[$6 \sqrt[5]{5}\$\]

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Introduction

Exponential expressions can be complex and challenging to simplify, but with the right approach, they can be broken down into manageable parts. In this article, we will explore how to simplify the expression (160⋅243)15(160 \cdot 243)^{\frac{1}{5}} and determine which of the given options is equal to it.

Understanding the Expression

The given expression is (160⋅243)15(160 \cdot 243)^{\frac{1}{5}}. To simplify this expression, we need to understand the properties of exponents and how to handle multiplication and division within the exponent.

Breaking Down the Expression

Let's start by breaking down the expression into smaller parts. We can rewrite the expression as (160⋅243)15=(16015⋅24315)15(160 \cdot 243)^{\frac{1}{5}} = (160^{\frac{1}{5}} \cdot 243^{\frac{1}{5}})^{\frac{1}{5}} using the property of exponents that states (a⋅b)n=an⋅bn(a \cdot b)^n = a^n \cdot b^n.

Simplifying the First Part

Now, let's simplify the first part of the expression, 16015160^{\frac{1}{5}}. We can rewrite 160160 as 53⋅425^3 \cdot 4^2. Using the property of exponents that states (am⋅bn)p=am⋅p⋅bn⋅p(a^m \cdot b^n)^p = a^{m \cdot p} \cdot b^{n \cdot p}, we can simplify the expression as follows:

16015=(53⋅42)15=53⋅15⋅425=535⋅425160^{\frac{1}{5}} = (5^3 \cdot 4^2)^{\frac{1}{5}} = 5^{3 \cdot \frac{1}{5}} \cdot 4^{\frac{2}{5}} = 5^{\frac{3}{5}} \cdot 4^{\frac{2}{5}}

Simplifying the Second Part

Next, let's simplify the second part of the expression, 24315243^{\frac{1}{5}}. We can rewrite 243243 as 353^5. Using the property of exponents that states (am)n=am⋅n(a^m)^n = a^{m \cdot n}, we can simplify the expression as follows:

24315=(35)15=35⋅15=31=3243^{\frac{1}{5}} = (3^5)^{\frac{1}{5}} = 3^{5 \cdot \frac{1}{5}} = 3^1 = 3

Combining the Parts

Now that we have simplified the first and second parts of the expression, we can combine them to get the final result. We have:

(160⋅243)15=(16015⋅24315)15=(535⋅425⋅3)15(160 \cdot 243)^{\frac{1}{5}} = (160^{\frac{1}{5}} \cdot 243^{\frac{1}{5}})^{\frac{1}{5}} = (5^{\frac{3}{5}} \cdot 4^{\frac{2}{5}} \cdot 3)^{\frac{1}{5}}

Simplifying the Final Expression

To simplify the final expression, we can use the property of exponents that states (a⋅b)n=an⋅bn(a \cdot b)^n = a^n \cdot b^n. We can rewrite the expression as follows:

(535⋅425⋅3)15=535⋅15⋅425⋅15⋅315=5325⋅4225⋅315(5^{\frac{3}{5}} \cdot 4^{\frac{2}{5}} \cdot 3)^{\frac{1}{5}} = 5^{\frac{3}{5} \cdot \frac{1}{5}} \cdot 4^{\frac{2}{5} \cdot \frac{1}{5}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 4^{\frac{2}{25}} \cdot 3^{\frac{1}{5}}

Evaluating the Options

Now that we have simplified the expression, we can evaluate the options to determine which one is equal to it. Let's examine each option:

  • Option A: 96. This option is not equal to the simplified expression.
  • Option B: 80. This option is not equal to the simplified expression.
  • Option C: 5555 \sqrt[5]{5}. This option is not equal to the simplified expression.
  • Option D: 6556 \sqrt[5]{5}. This option is not equal to the simplified expression.

Q&A: Simplifying Exponential Expressions

Q: What is the correct answer to the expression (160⋅243)15(160 \cdot 243)^{\frac{1}{5}}? A: To simplify the expression, we need to break it down into smaller parts and use the properties of exponents. We can rewrite the expression as (16015⋅24315)15(160^{\frac{1}{5}} \cdot 243^{\frac{1}{5}})^{\frac{1}{5}}.

Q: How do I simplify the first part of the expression, 16015160^{\frac{1}{5}}? A: We can rewrite 160160 as 53⋅425^3 \cdot 4^2. Using the property of exponents that states (am⋅bn)p=am⋅p⋅bn⋅p(a^m \cdot b^n)^p = a^{m \cdot p} \cdot b^{n \cdot p}, we can simplify the expression as follows:

16015=(53⋅42)15=53⋅15⋅425=535⋅425160^{\frac{1}{5}} = (5^3 \cdot 4^2)^{\frac{1}{5}} = 5^{3 \cdot \frac{1}{5}} \cdot 4^{\frac{2}{5}} = 5^{\frac{3}{5}} \cdot 4^{\frac{2}{5}}

Q: How do I simplify the second part of the expression, 24315243^{\frac{1}{5}}? A: We can rewrite 243243 as 353^5. Using the property of exponents that states (am)n=am⋅n(a^m)^n = a^{m \cdot n}, we can simplify the expression as follows:

24315=(35)15=35⋅15=31=3243^{\frac{1}{5}} = (3^5)^{\frac{1}{5}} = 3^{5 \cdot \frac{1}{5}} = 3^1 = 3

Q: How do I combine the parts to get the final result? A: Now that we have simplified the first and second parts of the expression, we can combine them to get the final result. We have:

(160⋅243)15=(16015⋅24315)15=(535⋅425⋅3)15(160 \cdot 243)^{\frac{1}{5}} = (160^{\frac{1}{5}} \cdot 243^{\frac{1}{5}})^{\frac{1}{5}} = (5^{\frac{3}{5}} \cdot 4^{\frac{2}{5}} \cdot 3)^{\frac{1}{5}}

Q: What is the final result of the expression? A: To simplify the final expression, we can use the property of exponents that states (a⋅b)n=an⋅bn(a \cdot b)^n = a^n \cdot b^n. We can rewrite the expression as follows:

(535⋅425⋅3)15=535⋅15⋅425⋅15⋅315=5325⋅4225⋅315(5^{\frac{3}{5}} \cdot 4^{\frac{2}{5}} \cdot 3)^{\frac{1}{5}} = 5^{\frac{3}{5} \cdot \frac{1}{5}} \cdot 4^{\frac{2}{5} \cdot \frac{1}{5}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 4^{\frac{2}{25}} \cdot 3^{\frac{1}{5}}

Q: Which of the given options is equal to the simplified expression? A: Let's examine each option:

  • Option A: 96. This option is not equal to the simplified expression.
  • Option B: 80. This option is not equal to the simplified expression.
  • Option C: 5555 \sqrt[5]{5}. This option is not equal to the simplified expression.
  • Option D: 6556 \sqrt[5]{5}. This option is not equal to the simplified expression.

However, we can rewrite the expression as $5^{\frac{3}{25}} \cdot 4^{\frac{2}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot (22){\frac{2}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\frac{4}{25}} \cdot 3^{\frac{1}{5}} = 5^{\frac{3}{25}} \cdot 2^{\