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?
To simplify 1,000^2/3, we first need to calculate the cube root of 1,000, which is 10.
Then, we raise the value of 10 to the power of 2, resulting in 10^2 = 10 * 10 = 100.
Therefore, Kali is correct. The correct value for 1,000^2/3 is 100.
To determine who has the correct simplification, let's break it down step by step:
The expression is 1,000^2/3.
Charles suggests that 1,000 = 10⋅10⋅10. According to him, 1,000^2/3 = 10.
Zayeer suggests that 1,000 = 10⋅10⋅10. According to him, 1,000^2/3 = 10⋅10⋅10 = 1,000.
Kali suggests that 1,000 = 10⋅10⋅10. According to her, 1,000^2/3 = 10⋅10 = 100.
Now let's verify who is correct by simplifying the expression step by step:
Step 1: Calculate 1,000^(2/3):
- To simplify this expression, we need to apply the exponent rules.
- The exponent rule for exponentiation of a fraction says we need to calculate the nth root of the base raised to the power of the numerator of the fraction.
- In this case, the numerator is 2, so we need to calculate the square root (2nd root) of the base 1,000.
Step 2: Calculate the 2nd root of 1,000:
- The 2nd root (or square root) of 1,000 is equal to √(1,000).
- By using a calculator, we find that √(1,000) ≈ 31.62.
Therefore, the correct simplification is 1,000^(2/3) ≈ 31.62.
None of the given options 10, 1,000, or 100 matches the correct value.
To determine the correct simplification, let's break down the steps taken by Charles, Zayeer, and Kali.
Charles claims that the correct simplification is 10 because 1,000 can be expressed as 10⋅10⋅10. However, to simplify 1,000^2/3, we need to raise each factor (10) to the power of 2/3. When we raise 10 to the power of 2/3, we get approximately 4.6416, not 10. Therefore, Charles's simplification of 10 is incorrect.
Zayeer suggests that the correct simplification is 1,000 because 1,000 can be expressed as 10⋅10⋅10. However, as mentioned earlier, we need to raise each factor (10) to the power of 2/3, not multiply them. Raising 10 to the power of 2/3 will give us approximately 4.6416. When we multiply this by the three 10s, we get approximately 13.8241. Therefore, Zayeer's simplification of 1,000 is also incorrect.
Kali argues that the correct simplification is 100 because 1,000 can be expressed as 10⋅10⋅10. Kali then correctly points out that raising 10 to the power of 2/3 results in approximately 4.6416. Therefore, when we multiply this by the two 10s (since we are working with cube roots), we get approximately 46.416. Consequently, Kali's simplification of 100 is incorrect as well.
In conclusion, none of the individuals have correctly simplified 1,000^2/3. The correct value, based on calculating 10^(2/3), is approximately 4.6416.