1Cademy - Simplifying $$\sqrt{98a^7b^5}$$

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Simplifying $\sqrt{63u^3v^5}$

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Simplifying $\sqrt{98a^7b^5}$

Simplify the square root $\sqrt{98a^7b^5}$ by extracting the largest perfect square factors.

Step 1 — Find the largest perfect square factor. For the coefficient 98, the largest perfect square factor is 49. For the variables $a^7$ and $b^5$ , the largest even powers are $a^6$ and $b^4$ . Together, the largest perfect square factor is $49a^6b^4$ . Rewrite the radicand as the product of this perfect square and the remaining factors $2ab$ : $\sqrt{98a^7b^5} = \sqrt{49a^6b^4 \cdot 2ab}$

Step 2 — Apply the Product Property. Separate the expression into two radicals: $\sqrt{49a^6b^4 \cdot 2ab} = \sqrt{49a^6b^4} \cdot \sqrt{2ab}$

Step 3 — Simplify. Simplify the perfect square root. Since $\sqrt{49} = 7$ , $\sqrt{a^6} = |a^3|$ (absolute value is required because an odd power is extracted from an even principal root), and $\sqrt{b^4} = b^2$ : $\sqrt{49a^6b^4} \cdot \sqrt{2ab} = 7|a^3|b^2\sqrt{2ab}$