To find the cube root of 32, we need to determine the number that, when multiplied by itself three times (or cubed), equals 32. Here is the step-by-step process:
1. Understand the cube root notation: The expression [tex]\(\sqrt[3]{32}\)[/tex] represents the cube root of 32, which is the number [tex]\(x\)[/tex] such that [tex]\(x^3 = 32\)[/tex].
2. Estimate an initial value: We recognize that [tex]\(2^3 = 8\)[/tex] and [tex]\(3^3 = 27\)[/tex]. Since [tex]\(8 < 32 < 27\)[/tex], the cube root of 32 should be between 2 and 3.
3. Narrow down the range: Let's refine our estimate by considering more precise values. We know that [tex]\(2.5^3 = 15.625\)[/tex] and [tex]\(3^3 = 27\)[/tex], and since [tex]\(15.625 < 32 < 27\)[/tex], the cube root of 32 is between 2.5 and 3.
4. Use a more precise approximation: We can refine our guess again. Trying 3.2, we see that:
[tex]\[
3.2^3 = 3.2 \times 3.2 \times 3.2 = 10.24 \times 3.2 = 32.768
\][/tex]
Since [tex]\(32.768\)[/tex] is close to [tex]\(32\)[/tex], we know the cube root is just slightly less than 3.2.
5. Converge to the exact value: To get even closer, additional computations or tools for more accurate values can be used, but after sufficient refinement:
[tex]\[
\sqrt[3]{32} \approx 3.1748021039363987
\][/tex]
Therefore, the cube root of 32 is approximately [tex]\(3.1748021039363987\)[/tex].
