Math / Algebra

QuestionSimplify. $\sqrt{63 v^{3}}$
Assume that the variable represents a positive real number.

Studdy Solution

STEP 1

What is this asking? We need to simplify the square root of 63 times v cubed, assuming v is positive. Watch out! Don't forget to simplify both the number and the variable under the square root!

STEP 2

1. Factor the number
2. Simplify the number's square root
3. Rewrite the variable's exponent
4. Simplify the variable's square root
5. Combine the simplified terms

STEP 3

Let's factor the number under the square root!
We're looking for perfect squares that are factors of $\sqrt{63}$ .
We can rewrite 63 as $9 \cdot 7$ .
So, we have $\sqrt{63} = \sqrt{9 \cdot 7}$ .
Nine is a perfect square, which is great!

STEP 4

Now, let's simplify the square root of the number.
Since $\sqrt{9} = 3$ , we can take the 3 out of the square root.
This gives us $3\sqrt{7}$ .

STEP 5

We can rewrite $v^3$ as $v^2 \cdot v$ .
This makes it easier to simplify the square root, because we know that $\sqrt{v^2}$ simplifies to v when v is positive.
So, $\sqrt{v^3} = \sqrt{v^2 \cdot v}$ .

STEP 6

Now, let's simplify the variable part of the square root.
Since $\sqrt{v^2} = v$ , we can take the v out of the square root.
This gives us $v\sqrt{v}$ .

STEP 7

Finally, let's combine everything!
We have $3\sqrt{7}$ and $v\sqrt{v}$ .
Multiplying these together gives us our simplified expression: $3v\sqrt{7v}$ .

STEP 8

$3v\sqrt{7v}$

Was this helpful?

QuestionSimplify. $\sqrt{63 v^{3}}$
Assume that the variable represents a positive real number.

Studdy Solution

STEP 1

What is this asking? We need to simplify the square root of 63 times v cubed, assuming v is positive. Watch out! Don't forget to simplify both the number and the variable under the square root!

STEP 2

1. Factor the number
2. Simplify the number's square root
3. Rewrite the variable's exponent
4. Simplify the variable's square root
5. Combine the simplified terms

STEP 3

Let's factor the number under the square root!
We're looking for perfect squares that are factors of $\sqrt{63}$ .
We can rewrite 63 as $9 \cdot 7$ .
So, we have $\sqrt{63} = \sqrt{9 \cdot 7}$ .
Nine is a perfect square, which is great!

STEP 4

Now, let's simplify the square root of the number.
Since $\sqrt{9} = 3$ , we can take the 3 out of the square root.
This gives us $3\sqrt{7}$ .

STEP 5

We can rewrite $v^3$ as $v^2 \cdot v$ .
This makes it easier to simplify the square root, because we know that $\sqrt{v^2}$ simplifies to v when v is positive.
So, $\sqrt{v^3} = \sqrt{v^2 \cdot v}$ .

STEP 6

Now, let's simplify the variable part of the square root.
Since $\sqrt{v^2} = v$ , we can take the v out of the square root.
This gives us $v\sqrt{v}$ .

STEP 7

Finally, let's combine everything!
We have $3\sqrt{7}$ and $v\sqrt{v}$ .
Multiplying these together gives us our simplified expression: $3v\sqrt{7v}$ .

STEP 8

$3v\sqrt{7v}$

Get Studdy

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.

QuestionSimplify. 63v3\sqrt{63 v^{3}}63v3​Assume that the variable represents a positive real number.

Studdy Solution

STEP 1

What is this asking? We need to simplify the square root of 63 times *v* cubed, assuming *v* is positive. Watch out! Don't forget to simplify both the number and the variable under the square root!

STEP 2

1. Factor the number2. Simplify the number's square root3. Rewrite the variable's exponent4. Simplify the variable's square root5. Combine the simplified terms

STEP 3

Let's **factor** the number under the square root!We're looking for **perfect squares** that are factors of 63\sqrt{63}63​.We can rewrite 63 as 9⋅79 \cdot 79⋅7.So, we have 63=9⋅7\sqrt{63} = \sqrt{9 \cdot 7}63​=9⋅7​.Nine is a perfect square, which is great!

STEP 4

Now, let's **simplify** the square root of the number.Since 9=3\sqrt{9} = 39​=3, we can take the 3 out of the square root.This gives us 373\sqrt{7}37​.

STEP 5

We can **rewrite** v3v^3v3 as v2⋅vv^2 \cdot vv2⋅v.This makes it easier to simplify the square root, because we know that v2\sqrt{v^2}v2​ simplifies to *v* when *v* is positive.So, v3=v2⋅v\sqrt{v^3} = \sqrt{v^2 \cdot v}v3​=v2⋅v​.

STEP 6

Now, let's **simplify** the variable part of the square root.Since v2=v\sqrt{v^2} = vv2​=v, we can take the *v* out of the square root.This gives us vvv\sqrt{v}vv​.

STEP 7

Finally, let's **combine** everything!We have 373\sqrt{7}37​ and vvv\sqrt{v}vv​.Multiplying these together gives us our **simplified expression**: 3v7v3v\sqrt{7v}3v7v​.

STEP 8

3v7v3v\sqrt{7v}3v7v​

Get Studdy

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.

QuestionSimplify. $\sqrt{63 v^{3}}$
Assume that the variable represents a positive real number.

What is this asking? We need to simplify the square root of 63 times v cubed, assuming v is positive. Watch out! Don't forget to simplify both the number and the variable under the square root!

1. Factor the number
2. Simplify the number's square root
3. Rewrite the variable's exponent
4. Simplify the variable's square root
5. Combine the simplified terms

Let's factor the number under the square root!
We're looking for perfect squares that are factors of $\sqrt{63}$ .
We can rewrite 63 as $9 \cdot 7$ .
So, we have $\sqrt{63} = \sqrt{9 \cdot 7}$ .
Nine is a perfect square, which is great!

Now, let's simplify the square root of the number.
Since $\sqrt{9} = 3$ , we can take the 3 out of the square root.
This gives us $3\sqrt{7}$ .

We can rewrite $v^3$ as $v^2 \cdot v$ .
This makes it easier to simplify the square root, because we know that $\sqrt{v^2}$ simplifies to v when v is positive.
So, $\sqrt{v^3} = \sqrt{v^2 \cdot v}$ .

Now, let's simplify the variable part of the square root.
Since $\sqrt{v^2} = v$ , we can take the v out of the square root.
This gives us $v\sqrt{v}$ .

Finally, let's combine everything!
We have $3\sqrt{7}$ and $v\sqrt{v}$ .
Multiplying these together gives us our simplified expression: $3v\sqrt{7v}$ .

$3v\sqrt{7v}$