Question

name
period __ date __
aim: how do we define and identify numbers and expressions as rational or irrational?
do now: pick your level and evaluate the value of each expression
\sqrt{49} =
is your response a fraction or integer?
yes □ no □
\sqrt{\frac{49}{196}} =
is your response a fraction or integer?
yes □ no □
\frac{\sqrt{144} + 200}{\sqrt{4}} =
is your response a fraction or integer?
yes □ no □
\frac{\sqrt{144} - \sqrt{80}}{2} =
is your response a fraction or integer?
yes □ no □

Explanation:

Step1: Evaluate $\sqrt{49}$

We know that $7\times7 = 49$, so $\sqrt{49}=7$.

Step2: Evaluate $\sqrt{\frac{49}{196}}$

First, simplify the fraction inside the square root: $\frac{49}{196}=\frac{1}{4}$. Then, $\sqrt{\frac{1}{4}}=\frac{1}{2}$ (since $(\frac{1}{2})^2=\frac{1}{4}$).

Step3: Evaluate $\frac{\sqrt{144} + 200}{\sqrt{4}}$

First, find the square roots: $\sqrt{144}=12$ and $\sqrt{4}=2$. Then substitute these values into the expression: $\frac{12 + 200}{2}=\frac{212}{2}=106$.

Step4: Evaluate $\frac{\sqrt{144}-\sqrt{80}}{2}$

First, find the square roots: $\sqrt{144}=12$ and $\sqrt{80}=4\sqrt{5}$ (since $80 = 16\times5$ and $\sqrt{16\times5}=4\sqrt{5}$). Then substitute into the expression: $\frac{12 - 4\sqrt{5}}{2}=6 - 2\sqrt{5}$.

For the "Is your response a fraction or integer?" questions:

• For $\sqrt{49}=7$, the response is an integer, so "Yes" (since 7 is an integer).
• For $\sqrt{\frac{49}{196}}=\frac{1}{2}$, the response is a fraction, so "Yes" (since $\frac{1}{2}$ is a fraction).
• For $\frac{\sqrt{144} + 200}{\sqrt{4}} = 106$, the response is an integer, so "Yes".
• For $\frac{\sqrt{144}-\sqrt{80}}{2}=6 - 2\sqrt{5}$, this is an irrational number (because of the $\sqrt{5}$ term), so it is neither a fraction nor an integer, so "No".

Answer:

• $\sqrt{49}=\boldsymbol{7}$ (Integer, Yes)
• $\sqrt{\frac{49}{196}}=\boldsymbol{\frac{1}{2}}$ (Fraction, Yes)
• $\frac{\sqrt{144} + 200}{\sqrt{4}}=\boldsymbol{106}$ (Integer, Yes)
• $\frac{\sqrt{144}-\sqrt{80}}{2}=\boldsymbol{6 - 2\sqrt{5}}$ (Neither, No)