Question

question 9
simplify
\\(\sqrt{\frac{10}{81}}\\)
\\(\frac{10}{9}\\)
\\(\frac{\sqrt{10}}{11}\\)
\\(9\sqrt{10}\\)
\\(\frac{\sqrt{10}}{9}\\)

question 10
simplify
\\(\sqrt{\frac{400}{5}}\\)

question 11
a student visiting the sears tower skydeck is 1353 feet above the ground. find the distance the student can see to the horizon. use the formula \\(d = \sqrt{1.5h}\\) to approximate the distance \\(d\\) in miles to the horizon when \\(h\\) is the height of the viewer’s eyes above the ground in feet. round to the nearest mile.

question 12
simplify the expression by rationalizing the denominator.
\\(\frac{4}{\sqrt{21}}\\)
\\(\frac{4\sqrt{21}}{}\\)

Explanation:

Question 9

Step1: Recall square root property

The square root of a fraction $\sqrt{\frac{a}{b}}$ can be written as $\frac{\sqrt{a}}{\sqrt{b}}$ (where $a\geq0$ and $b > 0$). For $\sqrt{\frac{10}{81}}$, we apply this property.
$\sqrt{\frac{10}{81}}=\frac{\sqrt{10}}{\sqrt{81}}$

Step2: Simplify the square root of the denominator

We know that $\sqrt{81} = 9$ since $9\times9 = 81$. So we substitute $\sqrt{81}$ with 9.
$\frac{\sqrt{10}}{\sqrt{81}}=\frac{\sqrt{10}}{9}$

Step1: Apply square root of fraction property

For $\sqrt{\frac{400}{5}}$, use the property $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$ ($a = 400$, $b = 5$).
$\sqrt{\frac{400}{5}}=\frac{\sqrt{400}}{\sqrt{5}}$

Step2: Simplify the square root of the numerator

Since $\sqrt{400}=20$ (because $20\times20 = 400$), we have:
$\frac{\sqrt{400}}{\sqrt{5}}=\frac{20}{\sqrt{5}}$

Step3: Rationalize the denominator

Multiply the numerator and denominator by $\sqrt{5}$ to rationalize the denominator.
$\frac{20}{\sqrt{5}}\times\frac{\sqrt{5}}{\sqrt{5}}=\frac{20\sqrt{5}}{5}$

Step4: Simplify the fraction

Divide 20 by 5: $\frac{20\sqrt{5}}{5}=4\sqrt{5}$

Step1: Identify the values

We are given the formula $d=\sqrt{1.5h}$, where $h = 1353$ feet.

Step2: Substitute the value of $h$ into the formula

Substitute $h = 1353$ into the formula: $d=\sqrt{1.5\times1353}$

Step3: Calculate the product inside the square root

First, calculate $1.5\times1353$. $1.5\times1353 = 2029.5$

Step4: Take the square root

Now, find $\sqrt{2029.5}$. $\sqrt{2029.5}\approx45.05$ (but we need to round to the nearest mile. Wait, actually, we made a mistake here. Wait, the formula is $d=\sqrt{1.5h}$ where $h$ is in feet and $d$ is in miles? Wait, no, actually, the correct formula for the distance to the horizon in miles when height $h$ is in feet is $d=\sqrt{\frac{3h}{20000}}$? Wait, no, the given formula is $d=\sqrt{1.5h}$ with $h$ in feet and $d$ in miles? Wait, let's recalculate. Wait, $1.5h$ when $h = 1353$ is $1.5\times1353=2029.5$. Then $\sqrt{2029.5}\approx45.05$. But that can't be right. Wait, maybe the formula is $d=\sqrt{1.5h}$ where $h$ is in feet and we need to convert feet to miles? Wait, no, the problem says "Use the formula $d=\sqrt{1.5h}$ to approximate the distance $d$ in miles to the horizon when $h$ is the height of the viewer's eyes above the ground in feet". So we proceed with the given formula. So $\sqrt{1.5\times1353}=\sqrt{2029.5}\approx45$ (when rounded to the nearest mile). Wait, $\sqrt{2029.5}\approx45.05$, so rounded to the nearest mile is 45.
Wait, let's check the calculation again. $1.5\times1353 = 2029.5$. $\sqrt{2029.5}\approx45.05$, so the nearest mile is 45.

Answer:

$\frac{\sqrt{10}}{9}$ (corresponding to the option $\boldsymbol{\frac{\sqrt{10}}{9}}$)

Question 10