Simplify $\sqrt{x^8y^3z^5}$

Key takeaway: This problem uses the rule that square roots of perfect squares come out of the radical. Break each variable into an even-power part and a leftover part.

Step 1: Factor each exponent into an even part and a remainder

Assuming $x,y,z\ge 0$:

$\sqrt{x^8y^3z^5}=\sqrt{x^8}\,\sqrt{y^3}\,\sqrt{z^5}$

Now simplify each factor:

• $\sqrt{x^8}=x^4$
• $\sqrt{y^3}=\sqrt{y^2\cdot y}=y\sqrt{y}$
• $\sqrt{z^5}=\sqrt{z^4\cdot z}=z^2\sqrt{z}$

Step 2: Multiply the results

$\sqrt{x^8y^3z^5}=x^4\cdot y\sqrt{y}\cdot z^2\sqrt{z}$

$= x^4yz^2\sqrt{yz}$

Final answer

$\boxed{x^4yz^2\sqrt{yz}}$

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Pitfalls the pros know 👇 A frequent mistake is to write $\sqrt{x^8}=x^8$. A square root cuts the exponent in half for perfect squares, so $\sqrt{x^8}=x^4$, not $x^8$.

What if the problem changes? If the variables were not assumed nonnegative, then even roots would require absolute values for the fully extracted even powers, such as $\sqrt{x^8}=|x^4|$. Under the usual assumption $x\ge 0$, this is simply $x^4$.

Tags: square root, perfect square, radical simplification