What number to the 3rd power equals 8?
The Correct Answer and Explanation is :
To find the number that, when raised to the third power, equals 8, we can express this mathematically as ( x^3 = 8 ). To solve for ( x ), we can take the cube root of both sides of the equation:
[
x = \sqrt[3]{8}
]
The cube root of 8 is 2, because:
[
2 \times 2 \times 2 = 8
]
Thus, the solution is ( x = 2 ).
Explanation
The concept of powers and roots is fundamental in mathematics. When we raise a number to a power, we multiply that number by itself a specified number of times. In this case, raising a number to the third power (or cubing it) means multiplying the number by itself twice more, for a total of three factors.
Cube roots, on the other hand, are the inverse operation of cubing. If we say that ( x^3 = y ), then ( x ) is the cube root of ( y ). The cube root of a number ( y ) is the number ( x ) that satisfies this equation. For instance, since ( 2^3 = 8 ), we can say ( \sqrt[3]{8} = 2 ).
In the context of cube roots, it’s important to note that every real number has one real cube root. This is different from square roots, where a positive number has two square roots (one positive and one negative). For example, while both ( 2 ) and ( -2 ) squared give ( 4 ), only ( 2 ) cubed results in ( 8 ).
Understanding cube roots is crucial in various fields, including algebra, geometry, and even in real-world applications such as calculating volumes. This knowledge helps us manipulate equations and solve problems that involve powers and roots effectively.