Math / Algebra

QuestionExpress the following in simplest radical form. All variables represent positive real numbers. (Simplify your answer completely.) $\frac{\sqrt{12 y^{3}}}{\sqrt{25 x}}$

Studdy Solution

STEP 1

What is this asking? Simplify a fraction with square roots and variables, making it look as neat and tidy as possible! Watch out! Don't forget to simplify both the numbers and the variables under the square roots!

STEP 2

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

STEP 3

Alright, let's tackle that numerator, $\sqrt{12y^3}$ !
We want to break it down into smaller, simpler pieces.
Remember, we're looking for perfect squares hiding inside!

STEP 4

We can rewrite $12$ as $4 \cdot 3$ , where $4$ is a perfect square!
And $y^3$ can be rewritten as $y^2 \cdot y$ , where $y^2$ is a perfect square!
So, $\sqrt{12y^3} = \sqrt{4 \cdot 3 \cdot y^2 \cdot y}$ .
See how we're strategically breaking things down?

STEP 5

Now, let's pull those perfect squares out! $\sqrt{4 \cdot 3 \cdot y^2 \cdot y} = \sqrt{4} \cdot \sqrt{y^2} \cdot \sqrt{3y} = 2y\sqrt{3y}$ .
Boom! Numerator simplified.

STEP 6

The denominator is $\sqrt{25x}$ . $25$ is a perfect square (it's $5^2$ ), so we can simplify this to $\sqrt{25} \cdot \sqrt{x} = 5\sqrt{x}$ .
Nice and clean!

STEP 7

Now, let's put it all together!
We have $\frac{2y\sqrt{3y}}{5\sqrt{x}}$ .
We've simplified the top and bottom separately, but we can make this even better!

STEP 8

To simplify further, we can multiply the top and bottom by $\sqrt{x}$ to get rid of the square root in the denominator.
This is called rationalizing the denominator.
So, we have $\frac{2y\sqrt{3y}}{5\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}} = \frac{2y\sqrt{3xy}}{5x}$ .

STEP 9

Our final simplified answer is $\frac{2y\sqrt{3xy}}{5x}$ !
We've simplified both the numbers and variables under the square roots and rationalized the denominator.
Excellent work!

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QuestionExpress the following in simplest radical form. All variables represent positive real numbers. (Simplify your answer completely.) $\frac{\sqrt{12 y^{3}}}{\sqrt{25 x}}$

Studdy Solution

STEP 1

What is this asking? Simplify a fraction with square roots and variables, making it look as neat and tidy as possible! Watch out! Don't forget to simplify both the numbers and the variables under the square roots!

STEP 2

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

STEP 3

Alright, let's tackle that numerator, $\sqrt{12y^3}$ !
We want to break it down into smaller, simpler pieces.
Remember, we're looking for perfect squares hiding inside!

STEP 4

We can rewrite $12$ as $4 \cdot 3$ , where $4$ is a perfect square!
And $y^3$ can be rewritten as $y^2 \cdot y$ , where $y^2$ is a perfect square!
So, $\sqrt{12y^3} = \sqrt{4 \cdot 3 \cdot y^2 \cdot y}$ .
See how we're strategically breaking things down?

STEP 5

Now, let's pull those perfect squares out! $\sqrt{4 \cdot 3 \cdot y^2 \cdot y} = \sqrt{4} \cdot \sqrt{y^2} \cdot \sqrt{3y} = 2y\sqrt{3y}$ .
Boom! Numerator simplified.

STEP 6

The denominator is $\sqrt{25x}$ . $25$ is a perfect square (it's $5^2$ ), so we can simplify this to $\sqrt{25} \cdot \sqrt{x} = 5\sqrt{x}$ .
Nice and clean!

STEP 7

Now, let's put it all together!
We have $\frac{2y\sqrt{3y}}{5\sqrt{x}}$ .
We've simplified the top and bottom separately, but we can make this even better!

STEP 8

To simplify further, we can multiply the top and bottom by $\sqrt{x}$ to get rid of the square root in the denominator.
This is called rationalizing the denominator.
So, we have $\frac{2y\sqrt{3y}}{5\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}} = \frac{2y\sqrt{3xy}}{5x}$ .

STEP 9

Our final simplified answer is $\frac{2y\sqrt{3xy}}{5x}$ !
We've simplified both the numbers and variables under the square roots and rationalized the denominator.
Excellent work!

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QuestionExpress the following in simplest radical form. All variables represent positive real numbers. (Simplify your answer completely.) 12y325x\frac{\sqrt{12 y^{3}}}{\sqrt{25 x}}25x​12y3​​

Studdy Solution

STEP 1

What is this asking? Simplify a fraction with square roots and variables, making it look as neat and tidy as possible! Watch out! Don't forget to simplify both the numbers *and* the variables under the square roots!

STEP 2

1. Simplify the numerator2. Simplify the denominator3. Simplify the fraction

STEP 3

Alright, let's **tackle** that numerator, 12y3\sqrt{12y^3}12y3​!We want to break it down into smaller, simpler pieces.Remember, we're looking for **perfect squares** hiding inside!

STEP 4

STEP 5

Now, let's pull those **perfect squares** out! 4⋅3⋅y2⋅y=4⋅y2⋅3y=2y3y\sqrt{4 \cdot 3 \cdot y^2 \cdot y} = \sqrt{4} \cdot \sqrt{y^2} \cdot \sqrt{3y} = 2y\sqrt{3y}4⋅3⋅y2⋅y​=4​⋅y2​⋅3y​=2y3y​.Boom! Numerator simplified.

STEP 6

The denominator is 25x\sqrt{25x}25x​. 252525 is a **perfect square** (it's 525^252), so we can simplify this to 25⋅x=5x\sqrt{25} \cdot \sqrt{x} = 5\sqrt{x}25​⋅x​=5x​.Nice and clean!

STEP 7

Now, let's put it all together!We have 2y3y5x\frac{2y\sqrt{3y}}{5\sqrt{x}}5x​2y3y​​.We've simplified the top and bottom separately, but we can make this even better!

STEP 8

STEP 9

Our **final simplified answer** is 2y3xy5x\frac{2y\sqrt{3xy}}{5x}5x2y3xy​​!We've simplified both the numbers and variables under the square roots and rationalized the denominator.Excellent work!

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionExpress the following in simplest radical form. All variables represent positive real numbers. (Simplify your answer completely.) $\frac{\sqrt{12 y^{3}}}{\sqrt{25 x}}$

What is this asking? Simplify a fraction with square roots and variables, making it look as neat and tidy as possible! Watch out! Don't forget to simplify both the numbers and the variables under the square roots!

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

Alright, let's tackle that numerator, $\sqrt{12y^3}$ !
We want to break it down into smaller, simpler pieces.
Remember, we're looking for perfect squares hiding inside!

Now, let's pull those perfect squares out! $\sqrt{4 \cdot 3 \cdot y^2 \cdot y} = \sqrt{4} \cdot \sqrt{y^2} \cdot \sqrt{3y} = 2y\sqrt{3y}$ .
Boom! Numerator simplified.

The denominator is $\sqrt{25x}$ . $25$ is a perfect square (it's $5^2$ ), so we can simplify this to $\sqrt{25} \cdot \sqrt{x} = 5\sqrt{x}$ .
Nice and clean!

Now, let's put it all together!
We have $\frac{2y\sqrt{3y}}{5\sqrt{x}}$ .
We've simplified the top and bottom separately, but we can make this even better!

Our final simplified answer is $\frac{2y\sqrt{3xy}}{5x}$ !
We've simplified both the numbers and variables under the square roots and rationalized the denominator.
Excellent work!