Rewrite the irrational cube root 189−−−√3 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 \(\sqrt[3]{189}\) as the product of an integer and another irrational cube root, we start by factoring the number 189.

First, we can find the prime factorization of 189:

1. Divide by 3: \[ 189 \div 3 = 63 \]
2. Divide 63 by 3: \[ 63 \div 3 = 21 \]
3. Divide 21 by 3: \[ 21 \div 3 = 7 \]
4. The number 7 is prime, so we stop here.

Therefore, the prime factorization of 189 is: \[ 189 = 3^3 \times 7 \]

Now we apply the cube root to this factorization: \[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7} \]

Using the property of cube roots \(\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}\), we can separate the cube roots: \[ \sqrt[3]{189} = \sqrt[3]{3^3} \times \sqrt[3]{7} \]

Now we can simplify \(\sqrt[3]{3^3}\): \[ \sqrt[3]{3^3} = 3 \]

Putting it all together, we obtain: \[ \sqrt[3]{189} = 3 \times \sqrt[3]{7} \]

Thus, the final result is: \[ \sqrt[3]{189} = 3\sqrt[3]{7} \]

Answer 2

make it a paragraph

Answer 3

To rewrite the irrational cube root \(\sqrt[3]{189}\) as the product of an integer and another irrational cube root, we first need to factor the number 189. The prime factorization of 189 can be determined by dividing by 3 repeatedly: \(189 \div 3 = 63\), then \(63 \div 3 = 21\), and finally \(21 \div 3 = 7\), where 7 is a prime number. This gives us the factorization \(189 = 3^3 \times 7\). Next, we apply the property of cube roots, which states that \(\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}\). Therefore, we can express \(\sqrt[3]{189}\) as \(\sqrt[3]{3^3 \times 7} = \sqrt[3]{3^3} \times \sqrt[3]{7}\). Simplifying \(\sqrt[3]{3^3}\) yields 3, resulting in the expression \(\sqrt[3]{189} = 3 \times \sqrt[3]{7}\). Thus, we conclude that \(\sqrt[3]{189}\) can be rewritten as \(3\sqrt[3]{7}\).

Answer 4

but now make it sound like a 7th grader

Answer 5

To rewrite the cube root of 189, or \(\sqrt[3]{189}\), as the product of a whole number and another cube root, we first need to break down 189 into its prime factors. We can start by dividing 189 by 3: \(189 \div 3 = 63\). Then, we divide 63 by 3 again: \(63 \div 3 = 21\). Finally, we divide 21 by 3 one more time: \(21 \div 3 = 7\). Since 7 is a prime number, we can stop there. So, we find that 189 can be written as \(3^3 \times 7\).

Now, we apply the property of cube roots that says the cube root of a product is the same as the product of the cube roots. This means we can write \(\sqrt[3]{189}\) as \(\sqrt[3]{3^3 \times 7} = \sqrt[3]{3^3} \times \sqrt[3]{7}\). We know that \(\sqrt[3]{3^3}\) simplifies to 3 because it’s like asking what number times itself three times gives 27, and that number is 3.

So, putting it all together, we get \(\sqrt[3]{189} = 3 \times \sqrt[3]{7}\). In simpler terms, we can say that the cube root of 189 is the same as 3 multiplied by the cube root of 7, or \(3\sqrt[3]{7}\).