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Rewrite the irrational cube root 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 (x) keyboard.

show the work short answer

5 answers

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

  1. Factor \( 189 \):

    • \( 189 = 3 \times 63 \)
    • \( 63 = 3 \times 21 \)
    • \( 21 = 3 \times 7 \)
    • Thus, \( 189 = 3^3 \times 7 \).

  2. Now we can express the cube root: \[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7}. \]

  3. Using the property of cube roots, we can separate it: \[ \sqrt[3]{189} = \sqrt[3]{3^3} \times \sqrt[3]{7}. \]

  4. Since \( \sqrt[3]{3^3} = 3 \), we now have: \[ \sqrt[3]{189} = 3 \times \sqrt[3]{7}. \]

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

easy

Glad you found it easy! If you have any more questions or need further assistance, feel free to ask!

make it easier

Sure! Here’s a simplified way to explain the steps for rewriting \( \sqrt[3]{189} \):

  1. Factor 189:
    \( 189 = 3 \times 63 \)
    \( 63 = 3 \times 21 \)
    \( 21 = 3 \times 7 \)
    So, \( 189 = 3^3 \times 7 \).

  2. Rewrite the cube root:
    \[ \sqrt[3]{189} = \sqrt[3]{3^3 \times 7} \]

  3. Split the cube root:
    \[ \sqrt[3]{189} = \sqrt[3]{3^3} \times \sqrt[3]{7} \]

  4. Simplify:
    Since \( \sqrt[3]{3^3} = 3 \), you get:
    \[ \sqrt[3]{189} = 3 \times \sqrt[3]{7} \]

Final Answer:
\[ \sqrt[3]{189} = 3 \times \sqrt[3]{7} \]

This keeps it straightforward and easy to understand! Let me know if you’d like more help!