problem 2\nsimplify:\n-5\\sqrt{12}-\\sqrt{27}\n\nproblem 5\nin the following right - triangle, find the side…

Problem 2

Explanation:

Step1: Simplify square - roots

Simplify $\sqrt{12}$ and $\sqrt{27}$. We know that $12 = 4\times3$ and $27=9\times3$, so $\sqrt{12}=\sqrt{4\times3}=2\sqrt{3}$ and $\sqrt{27}=\sqrt{9\times3}=3\sqrt{3}$. The expression $- 5\sqrt{12}-\sqrt{27}$ becomes $-5\times2\sqrt{3}-3\sqrt{3}$.

Step2: Perform multiplication

$-5\times2\sqrt{3}=-10\sqrt{3}$, so the expression is $-10\sqrt{3}-3\sqrt{3}$.

Step3: Combine like - terms

Combining the terms with $\sqrt{3}$, we get $(-10 - 3)\sqrt{3}=-13\sqrt{3}$.

Answer:

$-13\sqrt{3}$

Problem 5

Explanation:

Step1: Apply the Pythagorean theorem

In a right - triangle, the Pythagorean theorem is $a^{2}+b^{2}=c^{2}$, where $c$ is the hypotenuse and $a$ and $b$ are the legs. Here, if the hypotenuse is the side of length $11$ and the legs are $x$ and $7$, then $x^{2}+7^{2}=11^{2}$. So $x^{2}+49 = 121$.

Step2: Solve for $x^{2}$

Subtract 49 from both sides of the equation: $x^{2}=121 - 49=72$.

Step3: Solve for $x$

Take the square root of both sides. Since $x$ represents the length of a side of a triangle, we take the positive square root. $x=\sqrt{72}$. Simplify $\sqrt{72}=\sqrt{36\times2}=6\sqrt{2}\approx6\times1.414 = 8.48$.

Answer:

$8.48$