To insert three rational numbers between $\sqrt{\sqrt{3}}$ and $\sqrt{5}$, we can use the fact that there are rational numbers that are arbitrarily close to irrational numbers.
Let's first simplify the problem by finding a common denominator for $\sqrt{\sqrt{3}}$ and $\sqrt{5}$:
Now we want to find three rational numbers between $1.316$ and $2.236$. One way to do this is to find three fractions with a common denominator that lies between $1.316$ and $2.236$. Let's choose the denominator $1000$, which lies between $1316$ and $2236$. Then we can find the numerator for each fraction by choosing any integer between $1316$ and $2236$.
So, here are three rational numbers that lie between $\sqrt{\sqrt{3}}$ and $\sqrt{5}$:
$$\frac{1749}{1000} \approx 1.749$$
$$\frac{1977}{1000} \approx 1.977$$
$$\frac{2106}{1000} \approx 2.106$$
Note that these numbers are rational and lie strictly between $\sqrt{\sqrt{3}}$ and $\sqrt{5}$.
Answers & Comments
Step-by-step explanation:
To insert three rational numbers between $\sqrt{\sqrt{3}}$ and $\sqrt{5}$, we can use the fact that there are rational numbers that are arbitrarily close to irrational numbers.
Let's first simplify the problem by finding a common denominator for $\sqrt{\sqrt{3}}$ and $\sqrt{5}$:
$$\sqrt{\sqrt{3}} = (\sqrt{3})^{\frac{1}{4}} = 3^{\frac{1}{4}} \approx 1.316$$
$$\sqrt{5} = 5^{\frac{1}{2}} \approx 2.236$$
Now we want to find three rational numbers between $1.316$ and $2.236$. One way to do this is to find three fractions with a common denominator that lies between $1.316$ and $2.236$. Let's choose the denominator $1000$, which lies between $1316$ and $2236$. Then we can find the numerator for each fraction by choosing any integer between $1316$ and $2236$.
So, here are three rational numbers that lie between $\sqrt{\sqrt{3}}$ and $\sqrt{5}$:
$$\frac{1749}{1000} \approx 1.749$$
$$\frac{1977}{1000} \approx 1.977$$
$$\frac{2106}{1000} \approx 2.106$$
Note that these numbers are rational and lie strictly between $\sqrt{\sqrt{3}}$ and $\sqrt{5}$.