Question
Find the exact solution of x.
4x^3 = 500
^3√5
±5
5
-5
4x^3 = 500
^3√5
±5
5
-5
Answers
Answered by
GPT-4o mini
To find the exact solution for \( x \) in the equation \( 4x^3 = 500 \sqrt[3]{5} \), we can follow these steps:
1. Divide both sides by 4:
\[
x^3 = \frac{500 \sqrt[3]{5}}{4}
\]
Simplifying the right-hand side gives:
\[
x^3 = 125 \sqrt[3]{5} \quad \text{(since } \frac{500}{4} = 125 \text{)}
\]
2. To solve for \( x \), we will take the cube root of both sides:
\[
x = \sqrt[3]{125 \sqrt[3]{5}}
\]
3. Recognize that \( 125 = 5^3 \):
\[
x = \sqrt[3]{5^3 \cdot \sqrt[3]{5}}
\]
4. Rewrite \( \sqrt[3]{5} \) as \( 5^{1/3} \):
\[
x = \sqrt[3]{5^3 \cdot 5^{1/3}} = \sqrt[3]{5^{3 + 1/3}} = \sqrt[3]{5^{3 + 0.3333}} = \sqrt[3]{5^{10/3}}
\]
5. Thus,
\[
x = 5^{(10/3) / 3} = 5^{10/9}
\]
Hence, the exact solution for \( x \) is:
\[
x = 5^{10/9}
\]
This is not one of the answer choices provided. If the expected answer should be one of ±5 or simply 5, check to see if the constraints or additional information has affected what you were supposed to select. However, the exact solution based on the arithmetic is \( 5^{10/9} \).
1. Divide both sides by 4:
\[
x^3 = \frac{500 \sqrt[3]{5}}{4}
\]
Simplifying the right-hand side gives:
\[
x^3 = 125 \sqrt[3]{5} \quad \text{(since } \frac{500}{4} = 125 \text{)}
\]
2. To solve for \( x \), we will take the cube root of both sides:
\[
x = \sqrt[3]{125 \sqrt[3]{5}}
\]
3. Recognize that \( 125 = 5^3 \):
\[
x = \sqrt[3]{5^3 \cdot \sqrt[3]{5}}
\]
4. Rewrite \( \sqrt[3]{5} \) as \( 5^{1/3} \):
\[
x = \sqrt[3]{5^3 \cdot 5^{1/3}} = \sqrt[3]{5^{3 + 1/3}} = \sqrt[3]{5^{3 + 0.3333}} = \sqrt[3]{5^{10/3}}
\]
5. Thus,
\[
x = 5^{(10/3) / 3} = 5^{10/9}
\]
Hence, the exact solution for \( x \) is:
\[
x = 5^{10/9}
\]
This is not one of the answer choices provided. If the expected answer should be one of ±5 or simply 5, check to see if the constraints or additional information has affected what you were supposed to select. However, the exact solution based on the arithmetic is \( 5^{10/9} \).