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

1. Simplifying \( \sqrt{189} \): \[ 189 = 9 \times 21 = 3^2 \times 21 \] Now, we can take the square root: \[ \sqrt{189} = \sqrt{3^2 \times 21} = \sqrt{3^2} \times \sqrt{21} = 3\sqrt{21} \]

2. Substituting back into the cube root: Now we can substitute \( \sqrt{189} \) back into our original expression: \[ 3^{\sqrt{189}} = 3^{3\sqrt{21}} \]

3. Rewriting as a power: We can use properties of exponents to rewrite \( 3^{3\sqrt{21}} \): \[ 3^{3\sqrt{21}} = (3^3)^{\sqrt{21}} = 27^{\sqrt{21}} \]

4. Separate integer and irrational cube root: We can express this as: \[ 27^{\sqrt{21}} = 27 \cdot 3^{\sqrt{21} - 3} \]

5. Expressing the second term: Here, we recognize that this does not simplify cleanly; instead, we keep it as the infinite product involving cube roots.

However, since we need to express \( 3^{\sqrt{189}} \) more simply as the cube root term, we can directly express it: \[ 3^{\sqrt{189}} = 3^3 \cdot 3^{\sqrt{21}} = 27 \cdot 3^{\frac{1}{3} \cdot 21} = 27 \cdot \sqrt[3]{3^{21}} \]

So, we can write:

Final expression for \( 3^{\sqrt{189}} \): \[ 3^{\sqrt{189}} = 27 \sqrt[3]{3^{21}} = 27 \cdot 3^{7} \quad \textbf{Final:} \quad 27 \cdot \sqrt[3]{3^{21}} \]

In conclusion, the irrational cube root \( 3^{\sqrt{189}} \) can be expressed as: \[ 27 \cdot \sqrt[3]{3^{21}} \]

Answer 2

can you put it in plain language

Answer 3

Sure! Here’s a simplified explanation of how to rewrite the expression \( 3^{\sqrt{189}} \):

1. First, simplify \( \sqrt{189} \):

   • We can break down \( 189 \) into smaller factors: \( 189 = 9 \times 21 = 3^2 \times 21 \).
   • Then, the square root of \( 189 \) can be calculated: \[ \sqrt{189} = \sqrt{3^2 \times 21} = 3 \sqrt{21} \]

2. Substitute back into the expression:

   • Now, we replace \( \sqrt{189} \) in our original expression: \[ 3^{\sqrt{189}} = 3^{3 \sqrt{21}} \]

3. Use properties of exponents:

   • We can rewrite the expression using the fact that \( 3^{3\sqrt{21}} \) can be separated: \[ 3^{3 \sqrt{21}} = (3^3)^{\sqrt{21}} = 27^{\sqrt{21}} \]

4. Final expression:

   • So in simpler terms, we can say: \[ 3^{\sqrt{189}} = 27 \cdot 3^{\sqrt{21}} \]

In summary, the expression \( 3^{\sqrt{189}} \) can be rewritten as a product of the integer \( 27 \) and an irrational term involving \( 3^{\sqrt{21}} \).

Answer 4

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

Answer 5

Sure! Here’s the explanation without any special symbols:

1. First, simplify the square root of 189.

   • We can break down 189 into smaller factors: 189 equals 9 times 21, which can be written as 3 squared times 21.
   • Then, the square root of 189 can be calculated: the square root of 189 equals the square root of 3 squared times 21, which equals 3 times the square root of 21.

2. Substitute back into the expression.

   • Now we replace the square root of 189 in our original expression: three raised to the power of the square root of 189 equals three raised to the power of three times the square root of 21.

3. Use properties of exponents.

   • We can rewrite the expression using the fact that three raised to the power of three times the square root of 21 can be separated: three raised to the power of three times the square root of 21 equals 27 raised to the power of the square root of 21.

4. Final expression.

   • In simpler terms, we can say: three raised to the power of the square root of 189 equals 27 times three raised to the power of the square root of 21.

In summary, the expression three raised to the power of the square root of 189 can be rewritten as a product of the integer 27 and an irrational term involving three raised to the power of the square root of 21.

Answer 6

turn it into a paragraph and make it sound as an 8th grader

Answer 7

To simplify the expression three raised to the square root of 189, we first need to simplify the square root. We can break down 189 into smaller parts, which is 9 times 21, or 3 squared times 21. From this, we find that the square root of 189 is 3 times the square root of 21. Next, we substitute this back into our expression, so we have three raised to the square root of 189 as three raised to three times the square root of 21. Using some rules of exponents, we can rewrite this as 27 raised to the square root of 21. In simpler terms, we can say that three raised to the square root of 189 equals 27 times three raised to the square root of 21. So, in the end, we express this complex number as a combination of the whole number 27 and an irrational part involving three raised to the square root of 21.

Answer 8

put it in 4 sentences

Answer 9

To simplify the expression three raised to the square root of 189, we first find that the square root of 189 can be broken down into 3 times the square root of 21. Then, we substitute this back into our expression, giving us three raised to the power of three times the square root of 21. Using properties of exponents, we can rewrite this as 27 raised to the square root of 21. In simpler terms, three raised to the square root of 189 equals 27 times three raised to the square root of 21.

Answer 10

now make it sound as an 8th grader

Answer 11

To simplify the expression three raised to the square root of 189, we first need to figure out the square root of 189. We can break it down and find that the square root of 189 is the same as three times the square root of 21. Then, we can replace it in the original expression, so we have three raised to the power of three times the square root of 21. By using some rules of exponents, we can rewrite this as 27 times three raised to the power of the square root of 21.