Express in simplest radical form. $\newline$ $\frac{\sqrt{560}}{\sqrt{7}}$ $\newline$ Answer:

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Algebra 2

Simplify radical expressions with root inside the root

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Q. Express in simplest radical form.

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\frac{\sqrt{560}}{\sqrt{7}}

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Answer:

Factor and Simplify: Simplify the square root of $560$ . $\newline$ To simplify the square root of $560$ , we need to factor $560$ into its prime factors and look for pairs of the same number, since the square root of a pair of numbers is just that number. $\newline$ $560 = 2 \times 280$ $\newline$ $560 = 2 \times 2 \times 140$ $\newline$ $560 = 2 \times 2 \times 2 \times 70$ $\newline$ $560 = 2 \times 2 \times 2 \times 2 \times 35$ $\newline$ $560 = 2 \times 2 \times 2 \times 2 \times 5 \times 7$ $\newline$ Now we can pair the $2$ s and take them out of the square root. $\newline$ $\sqrt{560} = \sqrt{2 \times 2 \times 2 \times 2 \times 5 \times 7}$ $\newline$ $\sqrt{560} = 2 \times 2 \times \sqrt{5 \times 7}$ $\newline$ $\sqrt{560} = 4 \times \sqrt{35}$
Simplify Square Root of $7$ : Simplify the square root of $7$ . The square root of $7$ cannot be simplified further since $7$ is a prime number. $\sqrt{7} = \sqrt{7}$
Divide Simplified Square Roots: Divide the simplified square root of $560$ by the square root of $7$ . Now we divide $4 \times \sqrt{35}$ by $\sqrt{7}$ . $(4 \times \sqrt{35}) / (\sqrt{7})$ We can simplify this by dividing the square root of $35$ by the square root of $7$ . $(4 \times \sqrt{35}) / (\sqrt{7}) = 4 \times (\sqrt{35} / \sqrt{7})$
Simplify Division of Square Roots: Simplify the division of the square roots. $\newline$ We can simplify $\sqrt{35} / \sqrt{7}$ by recognizing that $\sqrt{35}$ is the same as $\sqrt{5 \times 7}$ and we can divide by $\sqrt{7}$ . $\newline$ $\sqrt{35} / \sqrt{7} = \sqrt{5 \times 7} / \sqrt{7}$ $\newline$ $\sqrt{35} / \sqrt{7} = \sqrt{5}$
Multiply by $4$ : Multiply the result by $4$ . $\newline$ Now we multiply the simplified square root by $4$ . $\newline$ $4 \times \sqrt{5}$ $\newline$ This is the simplest radical form of the original expression.