Math / Algebra

QuestionSimplify. $\sqrt{24 v^{11}}$
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 $24v^{11}$ , which means we're looking for a cleaner way to write this expression. Watch out! Remember that when simplifying square roots with variables, we need to consider whether the exponent is even or odd.

STEP 2

1. Factor the Number
2. Simplify the Numerical Square Root
3. Simplify the Variable Square Root
4. Combine and Celebrate

STEP 3

Let's break down that $24$ into its prime factors.
We're doing this to see which factors can be paired up to come out of the square root. $24$ can be factored as $2 \cdot 2 \cdot 2 \cdot 3$ , or $2^2 \cdot 2 \cdot 3$ .
So, we can rewrite our expression as $\sqrt{2^2 \cdot 2 \cdot 3 \cdot v^{11}}$ .

STEP 4

Now, look for pairs under the square root!
We have a pair of twos, $2^2$ .
Since $\sqrt{2^2} = 2$ , we can pull a 2 out.
This leaves us with $2\sqrt{2 \cdot 3 \cdot v^{11}}$ , which simplifies to $2\sqrt{6v^{11}}$ .

STEP 5

Alright, time to tackle that $v^{11}$ !
Remember, $v^{11}$ is the same as $v^{10} \cdot v^1$ .
Why did we do that?
Because $v^{10}$ has an even exponent, which is perfect for square roots!
We can rewrite $v^{10}$ as $(v^5)^2$ .
So, $\sqrt{v^{10}} = \sqrt{(v^5)^2} = v^5$ .

STEP 6

Now, we can pull that $v^5$ out of the square root.
This leaves us with $2v^5\sqrt{6v}$ .

STEP 7

Putting it all together, we get $2v^5\sqrt{6v}$ .
We've simplified the square root as much as possible!

STEP 8

$2v^5\sqrt{6v}$

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QuestionSimplify. $\sqrt{24 v^{11}}$
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 $24v^{11}$ , which means we're looking for a cleaner way to write this expression. Watch out! Remember that when simplifying square roots with variables, we need to consider whether the exponent is even or odd.

STEP 2

1. Factor the Number
2. Simplify the Numerical Square Root
3. Simplify the Variable Square Root
4. Combine and Celebrate

STEP 3

Let's break down that $24$ into its prime factors.
We're doing this to see which factors can be paired up to come out of the square root. $24$ can be factored as $2 \cdot 2 \cdot 2 \cdot 3$ , or $2^2 \cdot 2 \cdot 3$ .
So, we can rewrite our expression as $\sqrt{2^2 \cdot 2 \cdot 3 \cdot v^{11}}$ .

STEP 4

Now, look for pairs under the square root!
We have a pair of twos, $2^2$ .
Since $\sqrt{2^2} = 2$ , we can pull a 2 out.
This leaves us with $2\sqrt{2 \cdot 3 \cdot v^{11}}$ , which simplifies to $2\sqrt{6v^{11}}$ .

STEP 5

Alright, time to tackle that $v^{11}$ !
Remember, $v^{11}$ is the same as $v^{10} \cdot v^1$ .
Why did we do that?
Because $v^{10}$ has an even exponent, which is perfect for square roots!
We can rewrite $v^{10}$ as $(v^5)^2$ .
So, $\sqrt{v^{10}} = \sqrt{(v^5)^2} = v^5$ .

STEP 6

Now, we can pull that $v^5$ out of the square root.
This leaves us with $2v^5\sqrt{6v}$ .

STEP 7

Putting it all together, we get $2v^5\sqrt{6v}$ .
We've simplified the square root as much as possible!

STEP 8

$2v^5\sqrt{6v}$

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QuestionSimplify. 24v11\sqrt{24 v^{11}}24v11​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 24v1124v^{11}24v11, which means we're looking for a cleaner way to write this expression. Watch out! Remember that when simplifying square roots with variables, we need to consider whether the exponent is even or odd.

STEP 2

1. Factor the Number2. Simplify the Numerical Square Root3. Simplify the Variable Square Root4. Combine and Celebrate

STEP 3

STEP 4

Now, look for **pairs** under the square root!We have a pair of twos, 222^222.Since 22=2\sqrt{2^2} = 222​=2, we can pull a **2** out.This leaves us with 22⋅3⋅v112\sqrt{2 \cdot 3 \cdot v^{11}}22⋅3⋅v11​, which simplifies to 26v112\sqrt{6v^{11}}26v11​.

STEP 5

STEP 6

Now, we can pull that v5v^5v5 **out** of the square root.This leaves us with 2v56v2v^5\sqrt{6v}2v56v​.

STEP 7

Putting it all together, we get 2v56v2v^5\sqrt{6v}2v56v​.We've **simplified** the square root as much as possible!

STEP 8

2v56v2v^5\sqrt{6v}2v56v​

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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.

QuestionSimplify. $\sqrt{24 v^{11}}$
Assume that the variable represents a positive real number.

What is this asking? We need to simplify the square root of $24v^{11}$ , which means we're looking for a cleaner way to write this expression. Watch out! Remember that when simplifying square roots with variables, we need to consider whether the exponent is even or odd.

1. Factor the Number
2. Simplify the Numerical Square Root
3. Simplify the Variable Square Root
4. Combine and Celebrate

Now, look for pairs under the square root!
We have a pair of twos, $2^2$ .
Since $\sqrt{2^2} = 2$ , we can pull a 2 out.
This leaves us with $2\sqrt{2 \cdot 3 \cdot v^{11}}$ , which simplifies to $2\sqrt{6v^{11}}$ .

Now, we can pull that $v^5$ out of the square root.
This leaves us with $2v^5\sqrt{6v}$ .

Putting it all together, we get $2v^5\sqrt{6v}$ .
We've simplified the square root as much as possible!

$2v^5\sqrt{6v}$