Which expression is equivalent to the given expression?

Question

cube root of 40

Question 4Answer

a.
8 cube root of 5

b.
2 cube root of 5

c.
cube root of 10

d.
5 cube root of 2

Answer 1

To simplify the expression \( \sqrt[3]{40} \), we can break it down into its prime factors:

\[ 40 = 8 \times 5 = 2^3 \times 5 \]

Now, we can apply the property of cube roots:

\[ \sqrt[3]{40} = \sqrt[3]{8 \times 5} = \sqrt[3]{8} \times \sqrt[3]{5} \]

Since \( \sqrt[3]{8} = 2 \), we have:

\[ \sqrt[3]{40} = 2 \times \sqrt[3]{5} \]

Thus, \( \sqrt[3]{40} \) can be expressed as:

\[ 2 \sqrt[3]{5} \]

Looking at the provided options, the expression that corresponds to \( 2 \sqrt[3]{5} \) is:

b. \( 2 , \text{cube root of} , 5 \).

Answer 2

And why do you think that answer?

Answer 3

I arrived at the answer \( 2 , \text{cube root of} , 5 \) by simplifying the expression \( \sqrt[3]{40} \) into its constituent factors. Here’s a step-by-step clarification:

1. Understanding the Expression: The expression we started with is \( \sqrt[3]{40} \), which is asking for the cube root of 40.

2. Factoring 40: I broke down 40 into its prime factors: \[ 40 = 8 \times 5 = 2^3 \times 5 \] This is important because it allows us to express 40 in terms of perfect cubes and other factors.

3. Using the Properties of Cube Roots: The property of cube roots states: \[ \sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b} \] So when I applied this property, I separated \( \sqrt[3]{40} \) into: \[ \sqrt[3]{8 \times 5} = \sqrt[3]{8} \cdot \sqrt[3]{5} \]

4. Calculating the Cube Root: Since \( \sqrt[3]{8} = 2 \), I then had: \[ \sqrt[3]{40} = 2 \cdot \sqrt[3]{5} \]

5. Matching with Answer Choices: Finally, I looked at the given answer choices:

   • a. \( 8 , \text{cube root of} , 5 \)
   • b. \( 2 , \text{cube root of} , 5 \)
   • c. \( \text{cube root of} , 10 \)
   • d. \( 5 , \text{cube root of} , 2 \)

   The only option that matched my simplified expression \( 2 \sqrt[3]{5} \) was option b.

So, the reasoning is based on the mathematical properties of factoring and simplifying the cube root of a number, leading directly to the conclusion that option b is equivalent to the original expression \( \sqrt[3]{40} \).

Answer 4

no slashes please. or special characters

Answer 5

Sure! Let’s go through the explanation without slashes or special characters.

1. Understanding the Expression: We begin with the cube root of forty.

2. Factoring Forty: We can break down forty into its prime factors, which are eight and five. This can also be expressed as two to the power of three multiplied by five.

3. Using the Properties of Cube Roots: The cube root property states that the cube root of a product is equal to the product of the cube roots. Therefore, we separate the cube root of forty into the cube root of eight multiplied by the cube root of five.

4. Calculating the Cube Root: We know that the cube root of eight is two. So now we have two multiplied by the cube root of five.

5. Matching with Answer Choices: Looking at the answer choices provided:

   • Option a is eight multiplied by the cube root of five.
   • Option b is two multiplied by the cube root of five.
   • Option c is the cube root of ten.
   • Option d is five multiplied by the cube root of two.

   The option that matches our simplified expression of two multiplied by the cube root of five is option b.

Thus, the equivalent expression for the cube root of forty is option b: two multiplied by the cube root of five.