Definições de como calcular: Formas simplificadas

Simplest Form in Radical Expressions

Radicals, like fractions, can be simplified until you have the smallest possible parts. For starters, lets look at a simplified perfect square.

$$\sqrt{25} = 5$$

What is the square root of $25$? $5^2=25$, so we can simply $\sqrt{25}$ into $5$.

The same goes for $16$. Let's look at the square root of $16$.

$$\sqrt{16} = 4$$

This can also be done with numbers that aren't perfect squares. Let's look at a square root that will have some steps to simplifying.

$$\sqrt{18}$$

$18$ on its own isn't a perfect square, so we must find factors of $18$ that will give us a perfect square. We can use the multiplication and division properties of radicals to do that!

$$\begin{align} \sqrt{18}&=\sqrt{9 \cdot 2}\\ \sqrt{18}&=\sqrt{9}\sqrt{2}\\ \end{align}$$

We have factored $18$ into $9$ and $2$, so we can simplify $\sqrt{9}$ to get our final answer.

$$\sqrt{9}\sqrt{2} = 3\sqrt{2}$$

The simplest form of $\sqrt{18}$ is $3\sqrt{2}$.

Be mindful of what numbers are inside of the radical sign and which are not! Unless you can take the square root of the individual prime numbers, they must remain under the radical.

The same process works for variables under the radical. Let's start off from the basics again.

$$\sqrt{x^2}=x$$

$x \cdot x= x^2$, so $\sqrt{x^2}$ can be simplified into $x$. You can multiply different variables under the radical to simplify.

$$\sqrt{x^2 y^4 z}$$

First we're gonna separate the radical into parts to make it easier to work with, then we can simplify the parts.

$$\sqrt{x^2 y^4 z}\\ \sqrt{x^2}\sqrt{y^4}\sqrt{z}\\ x y^2 \sqrt{z}$$

The simplest form of $\sqrt{x^2 y^4 z}$ is $xy^2\sqrt{z}$. Notice that $\sqrt{z}$ can't be simplified any further in this case, but $x^2$ and $y^4$ are perfect squares.

Now what if the numbers under the radical aren't perfect squares? They can still be simplified, but there could still be something left under the radical, similar to the example with $\sqrt{18}$ in the first section.

$$\sqrt{x^5}$$

Let's take a look at this. First we need to factor it into parts that are perfect squares, but if we factor it into $x^2$ and $x^3$, we would have to simplify $\sqrt{x^3}$. The best way to simplify this would be to first factor it into $\sqrt{x^4}\sqrt{x}$.

$$\sqrt{x^5}\\ \sqrt{x^4}\sqrt{x}\\ x^2\sqrt{x}$$

Here's a practice problem! It combines all of the scenarios covered so far.

First, we can break these into separate radicals to make them easier to work with.

$$\sqrt{32} \sqrt{x^3} \sqrt{y^4} \sqrt{z^5}$$

Now lets simplify each of these before we combine them back into one expression.

$$\begin{align} \sqrt{32}&=\sqrt{16}\sqrt{2}\\ \sqrt{16}\sqrt{2} &= 4\sqrt{2}\\ \\ \sqrt{x^3}&=\sqrt{x^2}\sqrt{x}\\ \sqrt{x^2}\sqrt{x}&=x\sqrt{x}\\ \\ \sqrt{y^4}&=y^2\\ \\ \sqrt{z^5}&=\sqrt{z^4}\sqrt{z}\\ \sqrt{z^4}\sqrt{z}&=z^2\sqrt{z} \end{align}$$

Anything outside of the radical will stay outside of the radical, and everything inside will stay inside of the radical.

$$4\sqrt{2} \cdot x\sqrt{x} \cdot y^2 \cdot z^2 \sqrt{z}\\ 4x y^2 z^2 \sqrt{2xz}\\$$

The same process applies with other types of roots. Let's try an example with cubed roots.

$$\sqrt[3]{x^6 y^7}$$

We can factor $x^6$ into $x^2 \cdot x^2 \cdot x^2$ which means that $x^6$ is a perfect cube. Let's simplify that part in the expression before we move onto $y^7$.

$$\sqrt[3]{x^6y^7}=x^2\sqrt[3]{y^7}$$

$y^7$ can be factored into $y^2 \cdot y^2 \cdot y^2 \cdot y$. This means that we can pull the $y^2$ out of the radical, and leave the remaining $y$ under the radical.

$$x^2\sqrt[3]{y^7}=x^2y^2\sqrt[3]{x}$$

The simplest form of $\sqrt[3]{x^6 y^7}$ is $x^2y^2\sqrt[3]{x}$.

Here's one more problem to practice non-square roots.

(Video) Remove Variables from Radical Expressions

by mathman1024

This video by mathman1024 covers radical simplification.

$\large{\text{Summary}}$

We simplify a power under a radical by dividing it by the radical's index. For example,

$$\begin{align} \sqrt{x^{14}}&=\sqrt[2]{x^{14}}\\ &=x^{\frac{14}2}\\ &=x^7 \end{align}$$

Here, the exponent was $14$ and the index is $2$ (even though we don't usually write it when using the square root).

But what happens when the index doesn't divide evenly into the exponent? For example, with $\sqrt{x^{15}},$ the $2$ doesn't go into $15$ evenly ... we get $7.5$. So, we have to remove perfect squares from variables. You can rewrite the radical term as a product.

$$\sqrt{x^{15}}=\sqrt{x^{14+1}} =\sqrt{x^{14}\cdot x^1}$$

Now we have a perfect square term, $x^{14}$, which we can remove.

$$\begin{align} \sqrt{x^{15}}&=\sqrt{x^{14}\cdot x^1}\\ &=\sqrt{x^{14}}\cdot\sqrt{x}\\ &=x^7\sqrt{x} \end{align}$$

Simplifying Radical Expressions

Let's learn how to simplify radical expressions. You can remove constants with perfect factors, but variables change a little. You can use exponent rules to divide the power of an exponent by the index of the radical. The remainder of this division stays under the radical. Example $1$:

Simplify $\sqrt[4]{\left(810x^{12}y^{16}\right)}$

Radicals involving negative numbers are simplified using imaginary square roots.