The Correct Answer and Explanation is:
q = 5
The equation presented is q³ = 125. To solve for q, we need to understand what this equation represents. The term q³ signifies q raised to the third power, which is the mathematical operation of multiplying q by itself three times (q × q × q). The problem asks us to find the specific value of q that makes this statement true. This process involves finding the cube root of 125.
The term “cube root” has a direct connection to the geometry of a cube. If a cube has a side length of q, its volume is calculated as length times width times height, or q × q × q, which simplifies to q³. In the context of this problem, you can imagine we are given a cube with a volume of 125 cubic units, and we need to determine the length of one of its sides, which is q.
To find this value, we can search for a number that, when used as a factor three times, results in a product of 125. A good strategy is to test small, positive integers:
- 1³ = 1 × 1 × 1 = 1
- 2³ = 2 × 2 × 2 = 8
- 3³ = 3 × 3 × 3 = 27
- 4³ = 4 × 4 × 4 = 64
These results are all smaller than 125. Let’s test the next whole number, 5:
- 5³ = 5 × 5 × 5 = 125
This calculation provides the correct result. When we substitute 5 for q in the original equation, we get 5³ = 125, which is a true statement. Therefore, the solution to the equation is q = 5. The cube root of 125 is 5. This is the unique real number solution to the equation.
