Select The Correct Answer.Which Of The Following Is Equal To The Expression Below? ( 8 ⋅ 320 ) 1 3 (8 \cdot 320)^{\frac{1}{3}} ( 8 ⋅ 320 ) 3 1 ​ A. 40 B. 10 5 3 10 \sqrt[3]{5} 10 3 5 ​ C. 30 D. 8 5 3 8 \sqrt[3]{5} 8 3 5 ​

Introduction

Exponential expressions can be complex and challenging to simplify. However, with a clear understanding of the rules and properties of exponents, we can break down even the most daunting expressions into manageable parts. In this article, we will explore how to simplify the expression $(8 \cdot 320)^{\frac{1}{3}}$ and determine which of the given options is equal to it.

Understanding Exponents

Before we dive into the problem, let's review the basics of exponents. An exponent is a small number that is written to the right and above a larger number, indicating how many times the larger number should be multiplied by itself. For example, $2^3$ means $2$ multiplied by itself $3$ times, or $2 \cdot 2 \cdot 2$. When we have an exponent of $\frac{1}{3}$, it means we need to take the cube root of the expression.

Simplifying the Expression

Now that we have a solid understanding of exponents, let's simplify the expression $(8 \cdot 320)^{\frac{1}{3}}$. To do this, we need to follow the order of operations (PEMDAS):

1. Parentheses: Evaluate the expression inside the parentheses first. In this case, we have $8 \cdot 320$.
2. Exponents: Evaluate any exponents next. In this case, we have $\frac{1}{3}$.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Step 1: Evaluate the Expression Inside the Parentheses

Let's start by evaluating the expression inside the parentheses: $8 \cdot 320$. To do this, we simply multiply $8$ by $320$:

$$8 \cdot 320 = 2560$$

Step 2: Evaluate the Exponent

Now that we have the expression inside the parentheses evaluated, we can move on to the exponent. We have $\frac{1}{3}$, which means we need to take the cube root of the expression:

$$(8 \cdot 320)^{\frac{1}{3}} = \sqrt[3]{2560}$$

Step 3: Simplify the Cube Root

To simplify the cube root, we can try to find the cube root of the factors of $2560$. We can start by finding the prime factorization of $2560$:

$$2560 = 2^8 \cdot 5^1$$

Now that we have the prime factorization, we can take the cube root of each factor:

$$\sqrt[3]{2560} = \sqrt[3]{2^8 \cdot 5^1} = 2^{\frac{8}{3}} \cdot 5^{\frac{1}{3}}$$

Step 4: Simplify the Expression

Now that we have the cube root simplified, we can simplify the expression further. We can rewrite $2^{\frac{8}{3}}$ as $2^2 \cdot 2^{\frac{2}{3}}$:

$$2^{\frac{8}{3}} = 2^2 \cdot 2^{\frac{2}{3}} = 4 \cdot \sqrt[3]{2^2}$$

Now that we have the expression simplified, we can rewrite it in terms of the given options:

4 \cdot \sqrt[3]{2^2} = 4 \cdot \sqrt[3]{4} = 4 \cdot \sqrt[3]{2^2} = 4 \cdot \sqrt[3]{(2^2)^{\frac{1}{3}}} = 4 \cdot \sqrt[3]{2^{\frac{2}{3}}} = 4 \cdot \sqrt[3]{\frac{2^2}{2}} = 4 \cdot \sqrt[3]{\frac{4}{2}} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4<br/> **Q&A: Simplifying Exponential Expressions** =============================================

Q: What is the correct answer to the expression $(8 \cdot 320)^{\frac{1}{3}}$?

A: To simplify the expression, we need to follow the order of operations (PEMDAS). First, we evaluate the expression inside the parentheses: $8 \cdot 320 = 2560$. Then, we take the cube root of the expression: $\sqrt[3]{2560} = 2^{\frac{8}{3}} \cdot 5^{\frac{1}{3}}$. Finally, we simplify the expression further: $2^{\frac{8}{3}} = 4 \cdot \sqrt[3]{2^2}$.

Q: How do I simplify the cube root of a number?

A: To simplify the cube root of a number, we can try to find the cube root of the factors of the number. We can start by finding the prime factorization of the number. For example, if we want to simplify the cube root of $2560$, we can find its prime factorization: $2560 = 2^8 \cdot 5^1$. Then, we can take the cube root of each factor: $\sqrt[3]{2560} = \sqrt[3]{2^8 \cdot 5^1} = 2^{\frac{8}{3}} \cdot 5^{\frac{1}{3}}$.

Q: What is the difference between a cube root and a square root?

A: A cube root and a square root are both types of roots, but they differ in the power to which the number is raised. A square root is the number that, when multiplied by itself, gives the original number. For example, the square root of $16$ is $4$, because $4 \cdot 4 = 16$. A cube root, on the other hand, is the number that, when multiplied by itself twice, gives the original number. For example, the cube root of $64$ is $4$, because $4 \cdot 4 \cdot 4 = 64$.

Q: How do I evaluate an expression with a cube root?

A: To evaluate an expression with a cube root, we need to follow the order of operations (PEMDAS). First, we evaluate any expressions inside the parentheses. Then, we evaluate any exponents. Finally, we evaluate any multiplication and division operations from left to right.

Q: What is the correct answer to the expression $(8 \cdot 320)^{\frac{1}{3}}$ in terms of the given options?

A: To determine the correct answer, we need to simplify the expression further. We can rewrite $2^{\frac{8}{3}}$ as $2^2 \cdot 2^{\frac{2}{3}}$. Then, we can simplify the expression further: $2^2 \cdot 2^{\frac{2}{3}} = 4 \cdot \sqrt[3]{2^2} = 4 \cdot \sqrt[3]{4} = 4 \cdot \sqrt[3]{2^2} = 4 \cdot \sqrt[3]{(2²⁾{\frac{1}{3}}} = 4 \cdot \sqrt[3]{2^{\frac{2}{3}}} = 4 \cdot \sqrt[3]{\frac{2^2}{2}} = 4 \cdot \sqrt[3]{\frac{4}{2}} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} = 4 \cdot \sqrt[3]{2} =