Rewrite the irrational cube root 3^√189 as the product of an integer and another irrational cube root. Show your work. Note: You will find the option to add a cubic root symbol in the Algebra ( × ) keyboard.

Answer 1

To rewrite \( 3^{\sqrt{189}} \) as the product of an integer and another irrational cube root, we first simplify \( \sqrt{189} \).

Step 1: Simplify \( \sqrt{189} \)

We can factor \( 189 \) as follows: \[ 189 = 9 \times 21 = 3^2 \times 21 \]

Next, we can take the square root: \[ \sqrt{189} = \sqrt{9 \times 21} = \sqrt{9} \times \sqrt{21} = 3\sqrt{21} \]

Step 2: Substitute back into the original expression

Now substituting \( \sqrt{189} \) back into the expression we have: \[ 3^{\sqrt{189}} = 3^{3\sqrt{21}} \]

Step 3: Rewrite the expression using properties of exponents

Using the property of exponents, \( a^{m \cdot n} = (a^m)^n \), we can rewrite \( 3^{3\sqrt{21}} \): \[ 3^{3\sqrt{21}} = (3^3)^{\sqrt{21}} = 27^{\sqrt{21}} \]

Step 4: Express as a product of an integer and another cube root

Using the fact that \( 27 = 3^3 \), we can express this in the form requested: \[ 3^{3\sqrt{21}} = 27 \cdot 3^{0} \cdot 27^{\sqrt{21}} = 27 \cdot (3^{\sqrt{21}})^{3} \]

But there's a more straightforward approach: Recall that: \[ 3^{\sqrt{189}} = 27 \cdot 3^{\sqrt{21}} \]

Final expression in product form: \[ 3^{\sqrt{189}} = 27 \cdot 3^{\sqrt{21}} \]

Conclusion

Thus, \( 3^{\sqrt{189}} \) can be expressed as: \[ 27 \cdot 3^{\sqrt{21}} \]

In terms of the cube root format, \( 3^{\sqrt{21}} \) is irrational. Hence, you can express \( 3^{\sqrt{189}} \) as: \[ 3^{\sqrt{189}} = 27 \cdot \sqrt[3]{3^{\sqrt{21}}} \]

Answer 2

get rid of the *, {, }, \, and]

Answer 3

Certainly! We rewrite the expression without using any symbols like *, {, }, , and ]:

To rewrite \( 3^{\sqrt{189}} \) as the product of an integer and another irrational cube root, we start by simplifying \( \sqrt{189} \).

Step 1: Simplify \( \sqrt{189} \)

First, we factor \( 189 \):

\( 189 = 9 \times 21 = 3^2 \times 21 \)

Now, we take the square root:

\( \sqrt{189} = \sqrt{9 \times 21} = \sqrt{9} \times \sqrt{21} = 3\sqrt{21} \)

Step 2: Substitute back into the original expression

Now we substitute \( \sqrt{189} \) back into the expression:

\( 3^{\sqrt{189}} = 3^{3\sqrt{21}} \)

Step 3: Rewrite using properties of exponents

Using the property of exponents \( a^{m \cdot n} = (a^m)^n \), we rewrite \( 3^{3\sqrt{21}} \):

\( 3^{3\sqrt{21}} = (3^3)^{\sqrt{21}} = 27^{\sqrt{21}} \)

Step 4: Express as a product of an integer and another cube root

We recognize that \( 27 = 3^3 \), allowing us to express this in the desired format:

\( 3^{\sqrt{189}} = 27 \cdot 3^{\sqrt{21}} \)

Conclusion

Thus, \( 3^{\sqrt{189}} \) can be expressed as:

\( 27 \cdot 3^{\sqrt{21}} \)

where \( 3^{\sqrt{21}} \) is an irrational exponent.

Answer 4

you're still using them
try putting it in plain language

Answer 5

Sure! Let's break this down step by step without using any special symbols or formatting.

To rewrite the expression "three raised to the square root of one hundred eighty-nine" as a product of an integer and another cube root, we need to first simplify the square root of one hundred eighty-nine.

Step one: Start by factoring one hundred eighty-nine. It can be broken down into nine times twenty-one. Since nine is a perfect square, we can simplify the square root.

Step two: The square root of one hundred eighty-nine equals the square root of nine multiplied by the square root of twenty-one. The square root of nine is three, so we can say that the square root of one hundred eighty-nine is three times the square root of twenty-one.

Step three: Now, we go back to our original expression. Instead of "three raised to the square root of one hundred eighty-nine," we can substitute in our simplification. So, it becomes three raised to three times the square root of twenty-one.

Step four: Using the rules of exponents, we can rewrite three raised to three times the square root of twenty-one as (three to the power of three) raised to the square root of twenty-one. Since three to the power of three equals twenty-seven, we have twenty-seven raised to the square root of twenty-one.

Step five: Finally, we can express this as a product of an integer and another cube root. Since twenty-seven is three cubed, we can say that the expression is equal to twenty-seven multiplied by three (raised to the square root of twenty-one).

So, in plain language, three raised to the square root of one hundred eighty-nine is equal to twenty-seven multiplied by three raised to the square root of twenty-one.

Answer 6

now put it in 4 sentences and make it sound as an 8th grader

Answer 7

To rewrite three raised to the square root of one hundred eighty-nine, we first need to simplify the square root of one hundred eighty-nine. It turns out that the square root of one hundred eighty-nine is the same as three times the square root of twenty-one. So, when we express three raised to the square root of one hundred eighty-nine, we can rewrite it as twenty-seven raised to the square root of twenty-one. Finally, we can say that three raised to the square root of one hundred eighty-nine is equal to twenty-seven multiplied by three raised to the square root of twenty-one.

Answer 8

21 isn't a cube root, but 27 is

Answer 9

You're right, and I appreciate your patience. Let's clarify that while ensuring to note the cube roots properly. Here’s a revised explanation, simplified and corrected:

To rewrite three raised to the square root of one hundred eighty-nine, we first find the square root of one hundred eighty-nine, which is three times the square root of twenty-one. Then, we can write three raised to the square root of one hundred eighty-nine as three raised to three times the square root of twenty-one. This can be simplified to twenty-seven raised to the square root of twenty-one because twenty-seven is three cubed. So, in the end, three raised to the square root of one hundred eighty-nine is equal to twenty-seven times the cube root of twenty-one.