Math / Algebra

QuestionExpress $\frac{\sqrt{2}}{\sqrt{40}}$ as a fraction with a rational denominator. Give your answer in its simplest form.

Studdy Solution

STEP 1

What is this asking? We need to rewrite a fraction with a square root in the denominator, $\frac{\sqrt{2}}{\sqrt{40}}$ , so that the denominator becomes a nice, whole number, and then simplify it as much as possible! Watch out! Don't forget to simplify both the numerator and the denominator after you rationalize the denominator.
Look for common factors!

STEP 2

1. Simplify the denominator.
2. Rationalize the denominator.
3. Simplify the fraction.

STEP 3

Alright, let's simplify that $\sqrt{40}$ in the denominator!
We can rewrite $\sqrt{40}$ as $\sqrt{4 \cdot 10}$ .
Since $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ , we can rewrite this as $\sqrt{4} \cdot \sqrt{10}$ .
And since $\sqrt{4} = 2$ , we have $2\sqrt{10}$ .
So our fraction becomes $\frac{\sqrt{2}}{2\sqrt{10}}$ .
Much better!

STEP 4

Now, let's get rid of that pesky square root in the denominator!
To do this, we'll multiply both the numerator and the denominator by $\sqrt{10}$ .
Remember, multiplying by $\frac{\sqrt{10}}{\sqrt{10}}$ is the same as multiplying by one, so it doesn't change the value of our fraction, just how it looks!

STEP 5

Multiplying the numerator gives us $\sqrt{2} \cdot \sqrt{10} = \sqrt{2 \cdot 10} = \sqrt{20}$ .
Multiplying the denominator gives us $2\sqrt{10} \cdot \sqrt{10} = 2\sqrt{10 \cdot 10} = 2\sqrt{100} = 2 \cdot 10 = 20$ .
So, our fraction is now $\frac{\sqrt{20}}{20}$ .
Looking good!

STEP 6

We're almost there!
Let's simplify $\sqrt{20}$ .
We can rewrite $\sqrt{20}$ as $\sqrt{4 \cdot 5}$ , which is the same as $\sqrt{4} \cdot \sqrt{5}$ .
Since $\sqrt{4} = 2$ , we have $2\sqrt{5}$ .
So our fraction becomes $\frac{2\sqrt{5}}{20}$ .

STEP 7

Now, both the numerator and denominator have a factor of 2.
We can rewrite the fraction as $\frac{2 \cdot \sqrt{5}}{2 \cdot 10}$ .
Dividing both the numerator and the denominator by $2$ gives us $\frac{\sqrt{5}}{10}$ !

STEP 8

Our final, simplified answer is $\frac{\sqrt{5}}{10}$ !

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QuestionExpress $\frac{\sqrt{2}}{\sqrt{40}}$ as a fraction with a rational denominator. Give your answer in its simplest form.

Studdy Solution

STEP 1

STEP 2

1. Simplify the denominator.
2. Rationalize the denominator.
3. Simplify the fraction.

STEP 3

STEP 4

STEP 5

Multiplying the numerator gives us $\sqrt{2} \cdot \sqrt{10} = \sqrt{2 \cdot 10} = \sqrt{20}$ .
Multiplying the denominator gives us $2\sqrt{10} \cdot \sqrt{10} = 2\sqrt{10 \cdot 10} = 2\sqrt{100} = 2 \cdot 10 = 20$ .
So, our fraction is now $\frac{\sqrt{20}}{20}$ .
Looking good!

STEP 6

We're almost there!
Let's simplify $\sqrt{20}$ .
We can rewrite $\sqrt{20}$ as $\sqrt{4 \cdot 5}$ , which is the same as $\sqrt{4} \cdot \sqrt{5}$ .
Since $\sqrt{4} = 2$ , we have $2\sqrt{5}$ .
So our fraction becomes $\frac{2\sqrt{5}}{20}$ .

STEP 7

Now, both the numerator and denominator have a factor of 2.
We can rewrite the fraction as $\frac{2 \cdot \sqrt{5}}{2 \cdot 10}$ .
Dividing both the numerator and the denominator by $2$ gives us $\frac{\sqrt{5}}{10}$ !

STEP 8

Our final, simplified answer is $\frac{\sqrt{5}}{10}$ !

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QuestionExpress 240\frac{\sqrt{2}}{\sqrt{40}}40​2​​ as a fraction with a rational denominator. Give your answer in its simplest form.

Studdy Solution

STEP 1

STEP 2

1. Simplify the denominator.2. Rationalize the denominator.3. Simplify the fraction.

STEP 3

STEP 4

STEP 5

STEP 6

We're almost there!Let's **simplify** 20\sqrt{20}20​.We can rewrite 20\sqrt{20}20​ as 4⋅5\sqrt{4 \cdot 5}4⋅5​, which is the same as 4⋅5\sqrt{4} \cdot \sqrt{5}4​⋅5​.Since 4=2\sqrt{4} = 24​=2, we have 252\sqrt{5}25​.So our fraction becomes 2520\frac{2\sqrt{5}}{20}2025​​.

STEP 7

Now, both the numerator and denominator have a factor of **2**.We can rewrite the fraction as 2⋅52⋅10\frac{2 \cdot \sqrt{5}}{2 \cdot 10}2⋅102⋅5​​.Dividing both the numerator and the denominator by 222 gives us 510\frac{\sqrt{5}}{10}105​​!

STEP 8

Our final, simplified answer is 510\frac{\sqrt{5}}{10}105​​!

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QuestionExpress $\frac{\sqrt{2}}{\sqrt{40}}$ as a fraction with a rational denominator. Give your answer in its simplest form.

1. Simplify the denominator.
2. Rationalize the denominator.
3. Simplify the fraction.

We're almost there!
Let's simplify $\sqrt{20}$ .
We can rewrite $\sqrt{20}$ as $\sqrt{4 \cdot 5}$ , which is the same as $\sqrt{4} \cdot \sqrt{5}$ .
Since $\sqrt{4} = 2$ , we have $2\sqrt{5}$ .
So our fraction becomes $\frac{2\sqrt{5}}{20}$ .

Now, both the numerator and denominator have a factor of 2.
We can rewrite the fraction as $\frac{2 \cdot \sqrt{5}}{2 \cdot 10}$ .
Dividing both the numerator and the denominator by $2$ gives us $\frac{\sqrt{5}}{10}$ !

Our final, simplified answer is $\frac{\sqrt{5}}{10}$ !