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I am trying to simplify $7 \cdot\sqrt{5x}\cdot\sqrt{180x^5}$, given that $x$ is negative. The answer is $-210x^3$, but I am getting $210x^3$. Below is my reasoning:

For a $ k > 0 $, let $ k = - x $. Then, $7\cdot\sqrt{5x}\cdot\sqrt{180x^5}$ = $7\cdot5\cdot6\cdot i\sqrt{k} \cdot i\sqrt{k^5}$ = $210\cdot(-1)\cdot k^3$ = $ 210 \cdot (-1) \cdot k^3 = 210\cdot (-1) \cdot (-x)^3 = 210x^3$.

What am I doing wrong?

Any help is greatly appreciated

ə̷̶̸͇̘̜́̍͗̂̄︣͟
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ComputerScienceLearner
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  • $\sqrt{(-1)^2}\neq (\sqrt{-1})^2$. Your error is in combining $\sqrt{k}$ with $\sqrt{k^5}$ to get $k^3$. It can be corrected to be stated as $k=|x|$ and you'll have $k^3=|-x|^3=x^3$ – JMoravitz Dec 29 '17 at 16:15

2 Answers2

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Since $k=-x$, we have $$7\cdot\sqrt{5x}\cdot\sqrt{180x^5}=7\cdot i\cdot\sqrt{5k}\cdot i\sqrt{180k^5}=-7\cdot\sqrt{900k^6}=-7\cdot\sqrt{900x^6}=-210x^3$$

ə̷̶̸͇̘̜́̍͗̂̄︣͟
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$$\text{$7 \cdot\sqrt{5x}\cdot\sqrt{180x^5} \ $ where $ \ x<0$}$$

Lets let $x=-1$ to see what we should expect.

\begin{align} \left. \left( 7 \cdot\sqrt{5x}\cdot\sqrt{180x^5}\right)\right|_{x=-1} &= 7 \cdot\sqrt{-5}\cdot\sqrt{-180} \\ &= 7 \cdot i \cdot\sqrt{5}\cdot i \cdot \sqrt{180} \\ &= -7 \cdot \sqrt{900} \\ &= -210 \end{align}

Now Lets do it with $x=-n$ where $n > 0$.

\begin{align} \left. \left( 7 \cdot\sqrt{5x}\cdot\sqrt{180x^5}\right)\right|_{x=-n} &= 7 \cdot\sqrt{-5n}\cdot\sqrt{-180n^5} \\ &= 7 \cdot i \cdot\sqrt{5n}\cdot i \cdot \sqrt{180n^5} \\ &= -7 \cdot \sqrt{900n^6} \\ &= -210n^3 \\ &= -210(-x)^3 \\ &= 210x^3 \end{align}

Steven Alexis Gregory
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