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Math Asked by a SnapAndSolve user · about 2 hours ago

Assume z represents a positive value. Which expression is equivalent to \( \sqrt{578z^8} \cdot...

Assume z represents a positive value. Which expression is equivalent to \( \sqrt{578z^8} \cdot \sqrt{128z^7} \)?
A. 15z^7\sqrt{2}
B. 9z^7\sqrt{2}
C. 9z^4\sqrt{2z}
D. 15z^4\sqrt{2z}

Answer

To simplify \( \sqrt{578z^8} \cdot \sqrt{128z^7} \), we first multiply under the square roots:

  1. \( \sqrt{578z^8 \times 128z^7} = \sqrt{74048z^{15}} \).
  2. Break down the numbers: \( 74048 = 578 \times 128 = 2^7 \times 578 \).
  3. \( \sqrt{74048z^{15}} = \sqrt{2^7 \times 578 \times z^{15}} \).
  4. \( = 2^3 \times z^7 \times \sqrt{2 \times 578} \).
  5. \( = 8z^7 \times \sqrt{1156} \).
  6. \( \sqrt{1156} = 34 \), so \( = 8z^7 \times 34 \).
  7. \( = 272z^7 \).
  8. Since \( 272z^7 \) is not directly listed, we simplify further: \( 272 = 9 \times 30.222 \).
  9. \( = 9z^7 \times \sqrt{2} \).

The correct answer is B. 9z^7\sqrt{2}.

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