Charles, Zayeer, and Kali are trying to simplify 1,000 2/3 . Charles says the correct simplification is 10 because 1,000=10⋅10⋅10 and 1,000 2/3=10 . Zayeer says the correct simplification is 1,000 because 1,000=10⋅10⋅10→1,000 2/3=10⋅10⋅10=1,000 . Kali says the correct simplification is 100 because 1,000=10⋅10⋅10→1,000 2/3=10⋅10=100 . Who has the correct value?

Answer 1

To simplify \(1,000^{2/3}\), we start by expressing \(1,000\) in terms of its prime factorization. Since \(1,000 = 10^3\), we can rewrite the expression as follows:

\[ 1,000^{2/3} = (10^3)^{2/3} \]

Using the power of a power property of exponents, which states that \((a^m)^n = a^{m \cdot n}\), we can simplify:

\[ (10^3)^{2/3} = 10^{3 \cdot (2/3)} = 10^{2} = 100 \]

Thus, the correct simplification of \(1,000^{2/3}\) is \(100\).

Kali is correct in this case.