Question

question
express in simplest radical form.
$-5\sqrt{75} - \sqrt{147}$

Explanation:

Step1: Simplify each radical

First, factor the radicands (the numbers inside the square roots) into perfect squares and other factors.
For $\sqrt{75}$, we know that $75 = 25\times3$, and $25$ is a perfect square ($5^2$). So, $\sqrt{75}=\sqrt{25\times3}=\sqrt{25}\times\sqrt{3}=5\sqrt{3}$.
For $\sqrt{147}$, we factor $147 = 49\times3$, and $49$ is a perfect square ($7^2$). So, $\sqrt{147}=\sqrt{49\times3}=\sqrt{49}\times\sqrt{3}=7\sqrt{3}$.

Step2: Substitute the simplified radicals back into the original expression

The original expression is $-5\sqrt{75}-\sqrt{147}$. Substituting the simplified forms, we get:
$-5\times(5\sqrt{3}) - 7\sqrt{3}$

Step3: Perform the multiplication and combine like terms

First, multiply $-5$ and $5\sqrt{3}$: $-5\times5\sqrt{3}=-25\sqrt{3}$.
Now the expression is $-25\sqrt{3}-7\sqrt{3}$. Since both terms have $\sqrt{3}$, we can combine them by adding the coefficients: $(-25 - 7)\sqrt{3}=-32\sqrt{3}$.

Answer:

$-32\sqrt{3}$