Question

Use the duck principle to show that each equation is true. a. $\sqrt[4]{125} \cdot \sqrt[4]{5} = 5$ b. $\sqrt[3]{18} \cdot \sqrt[3]{12} = 6$ c. $\dfrac{\sqrt[5]{128}}{\sqrt[5]{16}} = \sqrt[5]{8}$ d. $\sqrt[4]{100} = \sqrt{10}$ e. $\sqrt[3]{4} \cdot \sqrt[6]{3} = \sqrt[6]{48}$

Solution

VerifiedStep 1

1 of 6$(a)\ \sqrt[4]{125}\ \cdot \sqrt[4]{5}$

From the rule that $\sqrt[4]{x}\ \cdot \sqrt[4]{y} = \sqrt[4]{xy}$

We get $\sqrt[4]{125}\ \cdot \sqrt[4]{5} = \sqrt[4]{625}$

Also $625 = 5^4$

So $5$ is the fourth root of 625.

Thus $\sqrt[4]{125}\ \cdot \sqrt[4]{5} = \sqrt[4]{625} = 5$

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