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

Key takeaway: The statement points out the main strategy: factor out the largest perfect square from each square root, then combine what remains.

Step 1: Use the product rule for square roots

a

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

Because the variables are assumed to be nonnegative, we can simplify directly.

Step 2: Extract the largest perfect square from each factor

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

Step 3: Combine the outside factors and inside factors

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

Final answer

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

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Pitfalls the pros know 👇 The main trap is stopping after simplifying only one factor. You need to simplify all three radicals and then combine the leftover radicals under one square root.

What if the problem changes? If one of the exponents were even, such as $z^4$ instead of $z^5$, then $\sqrt{z^4}=z^2$ and no $\sqrt{z}$ would remain. The leftover radical comes only from odd exponents.

Tags: perfect square, product rule, radical expression