PART 1
Henry explains why \(4^{\frac{3}{2}}=8\):
I know that \(4^3\) is \(64\) and the square root of \(64\) is \(8\).
Here is Henrietta's explanation for why \(4^{\frac{3}{2}}=8\):
I know that \(\sqrt{4}=2\) and the cube of \(2\) is \(8\).
PART 2
Given the following numbers, arrange them from least to greatest. Check your answers on the number line below.
\(A=4^{\frac{5}{2}}\) \(B=4^{\frac{3}{2}}\) \(C=25^{\frac{1}{2}}\) \(D=8^{\frac{2}{3}}\) \(E=8^{\frac{4}{3}}\) \(F=9^{\frac{3}{2}}\)
Henry explains why \(4^{\frac{3}{2}}=8\):
I know that \(4^3\) is \(64\) and the square root of \(64\) is \(8\).
Here is Henrietta's explanation for why \(4^{\frac{3}{2}}=8\):
I know that \(\sqrt{4}=2\) and the cube of \(2\) is \(8\).
- Are Henry and Henrietta correct? Explain.
- Calculate \(4^{\frac{5}{2}}\) and \(27^{\frac{2}{3}}\) using Henry's or Henrietta's strategy.
- Check your answers with a calculator.
PART 2
Given the following numbers, arrange them from least to greatest. Check your answers on the number line below.
\(A=4^{\frac{5}{2}}\) \(B=4^{\frac{3}{2}}\) \(C=25^{\frac{1}{2}}\) \(D=8^{\frac{2}{3}}\) \(E=8^{\frac{4}{3}}\) \(F=9^{\frac{3}{2}}\)
PART 3
Alicia and Zara are scientists working together. Alicia uses a calculator to evaluate \(3^{1.4}\) and gets an answer of \(6.473\). Zara thinks for a moment, makes some calculations on paper, and says "That cannot be right, because \(3^{1.4}\) must be less than \(6\)."
Find some hand calculations which show that, as Zara says, \(3^{1.4}\) must be less than \(6\).
(source: Illustrative Mathematics)