Question
On a test, Marcus is asked to simplify √128. Here is his work:
√128 = √16 · √8 = 4√8
Which statement best describes Marcus' answer?
√128 = √16 · √8 = 4√8
Which statement best describes Marcus' answer?
Answers
GPT-4o mini
Marcus' work is partially correct. He correctly identifies that \( \sqrt{128} \) can be simplified as \( \sqrt{16} \cdot \sqrt{8} \), and he correctly calculates \( \sqrt{16} = 4 \). However, he does not fully simplify \( \sqrt{8} \).
The complete simplification of \( \sqrt{8} \) is \( \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \). Therefore, substituting this back, we get:
\[
4\sqrt{8} = 4 \cdot 2\sqrt{2} = 8\sqrt{2}
\]
Thus, the best statement to describe Marcus' answer is that he made an error by not fully simplifying \( \sqrt{128} \). The correct simplification is \( 8\sqrt{2} \).
The complete simplification of \( \sqrt{8} \) is \( \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \). Therefore, substituting this back, we get:
\[
4\sqrt{8} = 4 \cdot 2\sqrt{2} = 8\sqrt{2}
\]
Thus, the best statement to describe Marcus' answer is that he made an error by not fully simplifying \( \sqrt{128} \). The correct simplification is \( 8\sqrt{2} \).