Question

what is the simplified form of $12x\sqrt{900x^2y^4z^6}$?
\bigcirc $30y^2|xz^3|$
\bigcirc $30x^2y^2z^4$
\bigcirc $360xy^2|xz^3|$
\bigcirc $360xy^2|z^3|$

Explanation:

Step1: Simplify the square root

First, simplify $\sqrt{900x^{2}y^{4}z^{6}}$. We know that $\sqrt{900} = 30$, $\sqrt{x^{2}}=\vert x\vert$, $\sqrt{y^{4}} = y^{2}$, and $\sqrt{z^{6}}=\vert z^{3}\vert$ (or $z^{3}$ if we consider non - negative, but with absolute value for generality). So $\sqrt{900x^{2}y^{4}z^{6}}=30\vert x\vert y^{2}\vert z^{3}\vert$.

Step2: Multiply by 12x

Now, multiply this by $12x$: $12x\times30\vert x\vert y^{2}\vert z^{3}\vert$. Calculate $12\times30 = 360$. So we get $360x\vert x\vert y^{2}\vert z^{3}\vert$. Using the property $a\vert a\vert=\vert a^{2}\vert=a^{2}$ (for real numbers, when we multiply $x$ and $\vert x\vert$, if $x\geq0$, $x\vert x\vert=x^{2}$; if $x < 0$, $x\vert x\vert=-x^{2}=\vert - x^{2}\vert$, but a more straightforward way is to rewrite $x\vert x\vert=\vert x^{2}\vert=x^{2}$ (since $x^{2}$ is non - negative). So $360x\vert x\vert y^{2}\vert z^{3}\vert=360x^{2}y^{2}\vert z^{3}\vert$? Wait, no, wait. Wait, the original expression is $12x\sqrt{900x^{2}y^{4}z^{6}}$. Wait, maybe I made a mistake in the first step. Let's re - do it.

Wait, $\sqrt{900x^{2}y^{4}z^{6}}=\sqrt{900}\times\sqrt{x^{2}}\times\sqrt{y^{4}}\times\sqrt{z^{6}} = 30\vert x\vert y^{2}\vert z^{3}\vert$. Then multiply by $12x$: $12x\times30\vert x\vert y^{2}\vert z^{3}\vert=360x\vert x\vert y^{2}\vert z^{3}\vert$. Now, $x\vert x\vert=\vert x^{2}\vert=x^{2}$ (because $x^{2}$ is always non - negative). So $360x\vert x\vert y^{2}\vert z^{3}\vert = 360x^{2}y^{2}\vert z^{3}\vert$? No, wait, looking at the options, one of the options is $360xy^{2}\vert xz^{3}\vert$. Wait, maybe we can rewrite $x\vert x\vert=\vert x^{2}\vert$ and then $x^{2}=\vert x\vert\times x$, so $360x\vert x\vert y^{2}\vert z^{3}\vert=360\vert x\vert\times x\times y^{2}\vert z^{3}\vert = 360xy^{2}\vert xz^{3}\vert$ (since $x\times\vert x\vert=\vert x^{2}\vert$ and $\vert a\vert\vert b\vert=\vert ab\vert$, so $x\vert x\vert=\vert x^{2}\vert$ and then $\vert x^{2}\vert\times\vert z^{3}\vert=\vert x^{2}z^{3}\vert$? No, wait, $x\vert x\vert=\vert x\vert\times x$, so $x\vert x\vert\vert z^{3}\vert=\vert x\vert\times x\times\vert z^{3}\vert=\vert x\vert\vert z^{3}\vert\times x=\vert xz^{3}\vert\times x$ (because $\vert a\vert\vert b\vert=\vert ab\vert$). So $360x\vert x\vert y^{2}\vert z^{3}\vert=360xy^{2}\vert xz^{3}\vert$.

Wait, let's start over. The expression is $12x\sqrt{900x^{2}y^{4}z^{6}}$.

First, simplify the square root:

$\sqrt{900x^{2}y^{4}z^{6}}=\sqrt{900}\times\sqrt{x^{2}}\times\sqrt{y^{4}}\times\sqrt{z^{6}}$

We know that:

• $\sqrt{900} = 30$
• $\sqrt{x^{2}}=\vert x\vert$
• $\sqrt{y^{4}}=y^{2}$ (since $y^{4}=(y^{2})^{2}$ and square root of a square is the absolute value, but $y^{2}$ is non - negative, so $\sqrt{y^{4}} = y^{2}$)
• $\sqrt{z^{6}}=\vert z^{3}\vert$ (since $z^{6}=(z^{3})^{2}$)

So $\sqrt{900x^{2}y^{4}z^{6}} = 30\vert x\vert y^{2}\vert z^{3}\vert$

Now, multiply by $12x$:

$12x\times30\vert x\vert y^{2}\vert z^{3}\vert=(12\times30)x\vert x\vert y^{2}\vert z^{3}\vert = 360x\vert x\vert y^{2}\vert z^{3}\vert$

Using the property $a\times\vert a\vert=\vert a^{2}\vert=a^{2}$ (for real numbers, when $a = x$, $x\vert x\vert=\vert x^{2}\vert=x^{2}$). But also, $x\vert x\vert\vert z^{3}\vert=\vert x\vert\times x\times\vert z^{3}\vert=\vert x\vert\vert z^{3}\vert\times x=\vert xz^{3}\vert\times x$ (because $\vert m\vert\vert n\vert=\vert mn\vert$). So $360x\vert x\vert y^{2}\vert z^{3}\vert=360xy^{2}\vert xz^{3}\vert$

Answer:

The simplified form is $360xy^{2}\vert xz^{3}\vert$ (the third option from the left, i.e., the option with $360xy^{2}\vert xz^{3}\vert$)