The solution to the equation x3=64 is called the cube root of 64.
The Correct Answer and Explanation is:
Here are the correct answers and the explanation for each expression.
Correct Answer:
a. 4
b. 5
c. -2
Explanation
The problem asks us to evaluate three different cube roots based on the provided definition. A cube root is the number that, when multiplied by itself three times, results in the number under the radical symbol. This is the inverse operation of cubing a number.
a. ³√64
To find the cube root of 64, we need to find a number, let’s call it x, such that x³ = 64. We are looking for the number that when used as a factor three times equals 64. We can test small integers to find the solution. We know 2 × 2 × 2 = 8, which is too small. Let’s try 3: 3 × 3 × 3 = 27, still too small. Let’s try 4: 4 × 4 = 16, and 16 × 4 = 64. Since 4³ = 64, the cube root of 64 is 4.
b. ³√125
Following the same logic, we are looking for a number that, when cubed, equals 125. From our previous calculation, we know 4³ is 64, so we should try the next whole number, which is 5. Let’s calculate 5³. The calculation is 5 × 5 × 5. First, 5 × 5 = 25. Then, 25 × 5 = 125. Because 5³ = 125, we have found that the cube root of 125 is 5.
c. ³√-8
This problem involves finding the cube root of a negative number. Unlike square roots, it is possible to find a real cube root for a negative number. The logic remains the same: we need a number x where x³ = -8. Since the result is negative, the number we are looking for must also be negative. A positive number cubed always yields a positive result. Let’s test small negative integers. We can try -2. The calculation would be (-2) × (-2) × (-2). First, (-2) × (-2) = 4. Then, we multiply this result by -2, so 4 × (-2) = -8. Since (-2)³ = -8, the cube root of -8 is -2.
