Carcass wrote:
If \(a^{\frac{2}{3}}=b^{\frac{2}{3}}\) for \(a ≠ 0\) and \(b ≠ 0\), then which of the following statements must be true?
Indicate \(all\) possible values.
\(\square\) \(\frac{a}{b}=1\)
\(\square\) \(\frac{a}{b}=-1\)
\(\square\) \((\frac{a}{b})^2=1\)
\(\square\) \(a=\frac{2}{3}\)
\(\square\) \(a^2=b^2\)
\(\square\) \(\sqrt{a}=\sqrt{b}\)
We have: \(a^{\frac{2}{3}}=b^{\frac{2}{3}}\)
Cubing both sides: \(a^{2}=b^{2}\) ... Option E is correct
Dividing throughout by \(b^2\): \(\frac{a^{2}}{b^{2}} = 1\) ... Option C is correct
Why are options A, B and F incorrect? We have already obtained: \(a^{2}=b^{2}\)
Taking square root: \(|a|=|b|\)
\(=> a = b\) or \(=> a = -b\)
However, we have to choose the options that
MUST be true (Note: Options A and B
may be true)
If either of \(a\) and \(b\) are negative (from above), their square root would become imaginary.Hence, option F is also incorrect
Answer: C and E