IDNLearn.com is your go-to resource for finding expert answers and community support. Get comprehensive answers to all your questions from our network of experienced experts.

Select the correct answer.

Which of the following is equal to this expression?

[tex]\[ (27 \cdot 250)^{\frac{1}{3}} \][/tex]

A. [tex]\( 15 \cdot \sqrt[3]{2} \)[/tex]

B. [tex]\( 5 \cdot \sqrt[3]{2} \)[/tex]

C. [tex]\( \sqrt[3]{10} \)[/tex]

D. [tex]\( \sqrt[3]{30} \)[/tex]


Sagot :

To solve the problem of determining which of the given choices is equal to the expression [tex]\((27 \cdot 250)^{\frac{1}{3}}\)[/tex], we analyze it step by step:

1. Evaluate the Original Expression:
First, we need to simplify the expression [tex]\((27 \cdot 250)^{\frac{1}{3}}\)[/tex].

2. Break Down the Expression:
[tex]\[ 27 \cdot 250 \][/tex]
Instead of calculating the product [tex]\(27 \cdot 250\)[/tex] directly, let's recognize that:
[tex]\[ 27 = 3^3 \quad \text{and} \quad 250 = 2 \cdot 125 = 2 \cdot 5^3 \][/tex]
Therefore,
[tex]\[ 27 \cdot 250 = 3^3 \cdot 2 \cdot 5^3 \][/tex]

3. Combine and Simplify Under the Cube Root:
Given the product inside the cube root, we have:
[tex]\[ (3^3 \cdot 2 \cdot 5^3)^{\frac{1}{3}} \][/tex]
Using the properties of exponents, we can split the terms within the cube root:
[tex]\[ (3^3)^{\frac{1}{3}} \cdot (2)^{\frac{1}{3}} \cdot (5^3)^{\frac{1}{3}} \][/tex]

4. Simplify Each Term:
Since [tex]\((a^b)^{\frac{1}{b}} = a\)[/tex], we know:
[tex]\[ (3^3)^{\frac{1}{3}} = 3 \][/tex]
Similarly,
[tex]\[ (5^3)^{\frac{1}{3}} = 5 \][/tex]
So we are left with:
[tex]\[ 3 \cdot 5 \cdot (2)^{\frac{1}{3}} \][/tex]
Simplify this further:
[tex]\[ 3 \cdot 5 = 15 \][/tex]
Thus, the expression is:
[tex]\[ 15 \cdot (2)^{\frac{1}{3}} \][/tex]

5. Compare with Given Choices:
[tex]\[ \boxed{15 \cdot \sqrt[3]{2}} \][/tex]
We see that this matches one of the given choices.

Therefore, the correct answer is:
[tex]\[ 15 \cdot \sqrt[3]{2} \][/tex]

Thus, the answer is [tex]\(\boxed{1}\)[/tex].