Math / Algebra

QuestionSimplify the radical expression. $\sqrt[3]{1000 x^{9}}$
Write your answer in the form $A, \sqrt[3]{B}$ , or $A \sqrt[3]{B}$ , where $A$ an in x . Use at most one radical in your answer, and at most expression for A .

Studdy Solution

STEP 1

What is this asking? We need to simplify a cube root containing a number and a variable, and write the answer in a specific format. Watch out! Remember the properties of radicals and exponents!
Don't forget to consider both the numerical and variable parts of the expression.

STEP 2

1. Simplify the numerical part
2. Simplify the variable part
3. Combine the simplified parts

STEP 3

We're looking at the cube root of $\sqrt[3]{1000}$ .
We want to find a number that, when multiplied by itself three times, equals 1000.
Let's think!

STEP 4

We know that $10 \cdot 10 \cdot 10 = 1000$ .
So, $\sqrt[3]{1000} = \textbf{10}$ .
Awesome!

STEP 5

Now, let's look at the cube root of $x^9$ , which is written as $\sqrt[3]{x^9}$ .
Remember, a cube root is the same as raising to the power of $\frac{1}{3}$ .

STEP 6

So, we have $(x^9)^{\frac{1}{3}}$ .
Using the power of a power rule, we multiply the exponents: $9 \cdot \frac{1}{3} = \frac{9}{3} = \textbf{3}$ .

STEP 7

Therefore, $\sqrt[3]{x^9} = x^3$ .
Perfect!

STEP 8

We found that $\sqrt[3]{1000} = 10$ and $\sqrt[3]{x^9} = x^3$ .

STEP 9

Since $\sqrt[3]{1000x^9}$ is the same as $\sqrt[3]{1000} \cdot \sqrt[3]{x^9}$ , we can multiply our simplified parts together.

STEP 10

So, our final simplified expression is $10 \cdot x^3 = 10x^3$ .
We've nailed it!

STEP 11

The simplified form of $\sqrt[3]{1000x^9}$ is $10x^3$ .
This fits the requested format where $A = 10$ and there's no radical part ( $B$ ).

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QuestionSimplify the radical expression. $\sqrt[3]{1000 x^{9}}$
Write your answer in the form $A, \sqrt[3]{B}$ , or $A \sqrt[3]{B}$ , where $A$ an in x . Use at most one radical in your answer, and at most expression for A .

Studdy Solution

STEP 1

What is this asking? We need to simplify a cube root containing a number and a variable, and write the answer in a specific format. Watch out! Remember the properties of radicals and exponents!
Don't forget to consider both the numerical and variable parts of the expression.

STEP 2

1. Simplify the numerical part
2. Simplify the variable part
3. Combine the simplified parts

STEP 3

We're looking at the cube root of $\sqrt[3]{1000}$ .
We want to find a number that, when multiplied by itself three times, equals 1000.
Let's think!

STEP 4

We know that $10 \cdot 10 \cdot 10 = 1000$ .
So, $\sqrt[3]{1000} = \textbf{10}$ .
Awesome!

STEP 5

Now, let's look at the cube root of $x^9$ , which is written as $\sqrt[3]{x^9}$ .
Remember, a cube root is the same as raising to the power of $\frac{1}{3}$ .

STEP 6

So, we have $(x^9)^{\frac{1}{3}}$ .
Using the power of a power rule, we multiply the exponents: $9 \cdot \frac{1}{3} = \frac{9}{3} = \textbf{3}$ .

STEP 7

Therefore, $\sqrt[3]{x^9} = x^3$ .
Perfect!

STEP 8

We found that $\sqrt[3]{1000} = 10$ and $\sqrt[3]{x^9} = x^3$ .

STEP 9

Since $\sqrt[3]{1000x^9}$ is the same as $\sqrt[3]{1000} \cdot \sqrt[3]{x^9}$ , we can multiply our simplified parts together.

STEP 10

So, our final simplified expression is $10 \cdot x^3 = 10x^3$ .
We've nailed it!

STEP 11

The simplified form of $\sqrt[3]{1000x^9}$ is $10x^3$ .
This fits the requested format where $A = 10$ and there's no radical part ( $B$ ).

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QuestionSimplify the radical expression. 1000x93\sqrt[3]{1000 x^{9}}31000x9​Write your answer in the form A,B3A, \sqrt[3]{B}A,3B​, or AB3A \sqrt[3]{B}A3B​, where AAA an in x . Use at most one radical in your answer, and at most expression for A .

Studdy Solution

STEP 1

What is this asking? We need to simplify a cube root containing a number and a variable, and write the answer in a specific format. Watch out! Remember the properties of radicals and exponents!Don't forget to consider both the numerical and variable parts of the expression.

STEP 2

1. Simplify the numerical part2. Simplify the variable part3. Combine the simplified parts

STEP 3

We're looking at the cube root of 10003\sqrt[3]{1000}31000​.We want to find a number that, when multiplied by itself three times, equals **1000**.Let's think!

STEP 4

We know that 10⋅10⋅10=100010 \cdot 10 \cdot 10 = 100010⋅10⋅10=1000.So, 10003=10\sqrt[3]{1000} = \textbf{10}31000​=10.Awesome!

STEP 5

Now, let's look at the cube root of x9x^9x9, which is written as x93\sqrt[3]{x^9}3x9​.Remember, a cube root is the same as raising to the power of 13\frac{1}{3}31​.

STEP 6

So, we have (x9)13(x^9)^{\frac{1}{3}}(x9)31​.Using the power of a power rule, we multiply the exponents: 9⋅13=93=39 \cdot \frac{1}{3} = \frac{9}{3} = \textbf{3}9⋅31​=39​=3.

STEP 7

Therefore, x93=x3\sqrt[3]{x^9} = x^33x9​=x3.Perfect!

STEP 8

We found that 10003=10\sqrt[3]{1000} = 1031000​=10 and x93=x3\sqrt[3]{x^9} = x^33x9​=x3.

STEP 9

Since 1000x93\sqrt[3]{1000x^9}31000x9​ is the same as 10003⋅x93\sqrt[3]{1000} \cdot \sqrt[3]{x^9}31000​⋅3x9​, we can multiply our simplified parts together.

STEP 10

So, our **final simplified expression** is 10⋅x3=10x310 \cdot x^3 = 10x^310⋅x3=10x3.We've nailed it!

STEP 11

The simplified form of 1000x93\sqrt[3]{1000x^9}31000x9​ is 10x310x^310x3.This fits the requested format where A=10A = 10A=10 and there's no radical part (BBB).

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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 the radical expression. $\sqrt[3]{1000 x^{9}}$
Write your answer in the form $A, \sqrt[3]{B}$ , or $A \sqrt[3]{B}$ , where $A$ an in x . Use at most one radical in your answer, and at most expression for A .

What is this asking? We need to simplify a cube root containing a number and a variable, and write the answer in a specific format. Watch out! Remember the properties of radicals and exponents!
Don't forget to consider both the numerical and variable parts of the expression.

1. Simplify the numerical part
2. Simplify the variable part
3. Combine the simplified parts

We're looking at the cube root of $\sqrt[3]{1000}$ .
We want to find a number that, when multiplied by itself three times, equals 1000.
Let's think!

We know that $10 \cdot 10 \cdot 10 = 1000$ .
So, $\sqrt[3]{1000} = \textbf{10}$ .
Awesome!

Now, let's look at the cube root of $x^9$ , which is written as $\sqrt[3]{x^9}$ .
Remember, a cube root is the same as raising to the power of $\frac{1}{3}$ .

So, we have $(x^9)^{\frac{1}{3}}$ .
Using the power of a power rule, we multiply the exponents: $9 \cdot \frac{1}{3} = \frac{9}{3} = \textbf{3}$ .

Therefore, $\sqrt[3]{x^9} = x^3$ .
Perfect!

We found that $\sqrt[3]{1000} = 10$ and $\sqrt[3]{x^9} = x^3$ .

Since $\sqrt[3]{1000x^9}$ is the same as $\sqrt[3]{1000} \cdot \sqrt[3]{x^9}$ , we can multiply our simplified parts together.

So, our final simplified expression is $10 \cdot x^3 = 10x^3$ .
We've nailed it!

The simplified form of $\sqrt[3]{1000x^9}$ is $10x^3$ .
This fits the requested format where $A = 10$ and there's no radical part ( $B$ ).