What is the exact solution to x
3
=150?
The Correct Answer and Explanation is :
To solve the equation ( x^3 = 150 ), we need to isolate ( x ). This can be accomplished by taking the cube root of both sides of the equation. The cube root is the inverse operation of cubing a number, meaning that if ( x^3 = a ), then ( x = \sqrt[3]{a} ).
Step-by-step Solution
- Identify the equation: We have ( x^3 = 150 ).
- Take the cube root: To find ( x ), we take the cube root of both sides:
[
x = \sqrt[3]{150}
] - Calculate the cube root: We can use a calculator or estimation techniques to find the cube root of 150. The value of ( \sqrt[3]{150} ) is approximately 5.313.
Explanation of the Cube Root
The cube root of a number ( a ) is a value ( x ) such that when multiplied by itself three times (cubed), it equals ( a ):
[
x^3 = a
]
In this case, finding the cube root of 150 involves determining which number, when raised to the third power, results in 150.
Cube roots can also be understood in a geometric context. If you visualize a cube with a volume of 150 cubic units, the side length of that cube would be ( \sqrt[3]{150} ). This concept is useful in various fields, including geometry, engineering, and physics.
In practical terms, cube roots are common in various applications, such as calculating dimensions of objects with a known volume, solving certain equations in algebra, and in various scientific calculations where dimensions must be considered.
Thus, the exact solution to the equation ( x^3 = 150 ) is:
[
x = \sqrt[3]{150} \approx 5.313
]
This value is important for further calculations and applications where the dimensions related to the volume of a cube are needed.