Simplify $7$ times the square root of $8$ minus five times the square root of $80$ plus two times the square root if $180$ plus the square root of $50$

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Math Problems

Grade 6

Estimate products of mixed numbers

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Q. Simplify

7

times the square root of

8

minus five times the square root of

80

plus two times the square root if

180

plus the square root of

50

Recognize Perfect Squares: Simplify each square root by factoring out perfect squares. $\newline$ $\sqrt{8}$ can be simplified by recognizing that $8 = 4 \times 2$ , where $4$ is a perfect square. $\newline$ $\sqrt{8} = \sqrt{(4 \times 2)} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}$
Factor Out Perfect Squares: Simplify $\sqrt{80}$ by factoring out perfect squares. $\sqrt{80}$ can be simplified by recognizing that $80 = 16 \times 5$ , where $16$ is a perfect square. $\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4\sqrt{5}$
Substitute Simplified Square Roots: Simplify $\sqrt{180}$ by factoring out perfect squares. $\newline$ $\sqrt{180}$ can be simplified by recognizing that $180 = 36 \times 5$ , where $36$ is a perfect square. $\newline$ $\sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \times \sqrt{5} = 6\sqrt{5}$
Combine Like Terms: Simplify $\sqrt{50}$ by factoring out perfect squares. $\newline$ $\sqrt{50}$ can be simplified by recognizing that $50 = 25 \times 2$ , where $25$ is a perfect square. $\newline$ $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$
Write Final Expression: Substitute the simplified square roots back into the original expression. $\newline$ $7\sqrt{8}$ becomes $7 \times 2\sqrt{2}$ , which is $14\sqrt{2}$ . $\newline$ $-5\sqrt{80}$ becomes $-5 \times 4\sqrt{5}$ , which is $-20\sqrt{5}$ . $\newline$ $2\sqrt{180}$ becomes $2 \times 6\sqrt{5}$ , which is $12\sqrt{5}$ . $\newline$ $\sqrt{50}$ becomes $7 \times 2\sqrt{2}$ $0$ . $\newline$ The expression now looks like this: $7 \times 2\sqrt{2}$ $1$
Write Final Expression: Substitute the simplified square roots back into the original expression. $\newline$ $7\sqrt{8}$ becomes $7 \times 2\sqrt{2}$ , which is $14\sqrt{2}$ . $\newline$ $-5\sqrt{80}$ becomes $-5 \times 4\sqrt{5}$ , which is $-20\sqrt{5}$ . $\newline$ $2\sqrt{180}$ becomes $2 \times 6\sqrt{5}$ , which is $12\sqrt{5}$ . $\newline$ $\sqrt{50}$ becomes $7 \times 2\sqrt{2}$ $0$ . $\newline$ The expression now looks like this: $7 \times 2\sqrt{2}$ $1$ Combine like terms. $\newline$ $14\sqrt{2}$ and $7 \times 2\sqrt{2}$ $0$ are like terms, as are $-20\sqrt{5}$ and $12\sqrt{5}$ . $\newline$ Combine $14\sqrt{2}$ and $7 \times 2\sqrt{2}$ $0$ to get $7 \times 2\sqrt{2}$ $8$ . $\newline$ Combine $-20\sqrt{5}$ and $12\sqrt{5}$ to get $14\sqrt{2}$ $1$ . $\newline$ The expression now looks like this: $14\sqrt{2}$ $2$
Write Final Expression: Substitute the simplified square roots back into the original expression. $\newline$ $7\sqrt{8}$ becomes $7 \times 2\sqrt{2}$ , which is $14\sqrt{2}$ . $\newline$ $-5\sqrt{80}$ becomes $-5 \times 4\sqrt{5}$ , which is $-20\sqrt{5}$ . $\newline$ $2\sqrt{180}$ becomes $2 \times 6\sqrt{5}$ , which is $12\sqrt{5}$ . $\newline$ $\sqrt{50}$ becomes $7 \times 2\sqrt{2}$ $0$ . $\newline$ The expression now looks like this: $7 \times 2\sqrt{2}$ $1$ Combine like terms. $\newline$ $14\sqrt{2}$ and $7 \times 2\sqrt{2}$ $0$ are like terms, as are $-20\sqrt{5}$ and $12\sqrt{5}$ . $\newline$ Combine $14\sqrt{2}$ and $7 \times 2\sqrt{2}$ $0$ to get $7 \times 2\sqrt{2}$ $8$ . $\newline$ Combine $-20\sqrt{5}$ and $12\sqrt{5}$ to get $14\sqrt{2}$ $1$ . $\newline$ The expression now looks like this: $14\sqrt{2}$ $2$ Write the final simplified expression. $\newline$ The final simplified expression is $14\sqrt{2}$ $2$ .