减根数表达式

Subtracting Radical Terms

Adding and subtracting radicals is really helpful to know when solving more complex algebraic problems, especially those with radical signs.

Radical terms (also known as roots) can be subtracted just like integers. However, subtracting radicals might not be simplified in certain cases because you can only subtract ‘like’ radical terms.

You can also find square roots with sums or differences.

If you have the same number under the same radical/root, you can subtract the radicals. Radicals are similar to variables. You can subtract like variables but you cannot subtract unlike variables.

Let’s take a look at an example. Simplify $4\sqrt{5}- \sqrt{5}$.

These two terms have the same number under the radical, so we can subtract the terms and simplify.

$$\begin{align} 4\sqrt{5}- \sqrt{5} &= 4\sqrt{5} - 1\sqrt{5} \\ &= (4-1) \sqrt{5} \\ &= 3 \sqrt{5} \\ \end{align}$$

Now, let’s simplify $9\sqrt{13}- 7\sqrt{2}-5\sqrt{13}$

We can only combine ‘like’ radicals/roots. The first and last term can be combined, but not the middle term.

Let's look at some problems.

$$\begin{align} 21\sqrt{17}- 8\sqrt{17} &= (21-8) \sqrt{17}\\ &= 13\sqrt{17}\\ \end{align}$$

$$\begin{align} 7\sqrt{19}+10\sqrt{3}-5\sqrt{3}-6\sqrt{19}&=7\sqrt{19}-6\sqrt{19}+10\sqrt{3}-5\sqrt{3} \\ &= (7-6) \sqrt{19}+ (10-5) \sqrt{3} \\ &= 1\sqrt{19} +5\sqrt{3} \\ &= \sqrt{19} +5\sqrt{3} \\ \end{align}$$

$$\begin{align} 2\sqrt{16}- \sqrt{9} &= 2 \sqrt{4^2} -\sqrt{3^2} \\ &= 2 \cdot 4 - 3 \\ &= 8-3 \\ &=5\\ \end{align}$$

$$\begin{align} \sqrt{27} -3\sqrt{45}-4\sqrt{3}&= \sqrt{3 \cdot 3 \cdot 3}-3 \sqrt{3 \cdot 3 \cdot 5} -4 \sqrt{3} \\ &= 3\sqrt{3}-(3 \cdot 3)\sqrt{5}-4\sqrt{3} \\ &= 3\sqrt{3}-4\sqrt{3} -9\sqrt{5}\\ &= (3-4) \sqrt{3}-9\sqrt{5} \\ &= -1\sqrt{3}-9\sqrt{5} \\ &= -\sqrt{3} -9\sqrt{5} \end{align}$$

Subtracting Radical Terms

Radicals work like variables when you add and subtract them. For example, if we have the expression,

$$2x - 8y - 4x$$

we can simplify it to by combining like terms:

$$-2x - 8y$$

because we can combine like terms. Radicals work the same way. We can substitute $x$ with $\sqrt{31}$ and $y$ with $\sqrt{94}$:

$$\begin{align} &2\sqrt{31} - 8\sqrt{94} - 4\sqrt{31}\\ &= -2\sqrt{31} - 8\sqrt{94} \end{align}$$

Notice that you can only subtract radical terms if the same number is under the same degree of radical!!

$$5\sqrt{10} - 7\sqrt{65} - 2\sqrt{65} = 5\sqrt{10} - 9\sqrt{65}$$

$\sqrt{65}$ is multiplied by $7$ in one term and $2$ in another. We can combine them by subtracting $2$ from $-7$ to get $-9$.

A good way to double check your work is to convert all monomials to decimals and determine whether the values are equal.

(Video) Add and Subtract Radicals

by Allie Cunningham

This video by Allie Cunningham explains radical subtraction.

$\large{\text{Summary}}$

In order to subtract two terms with radicals, both radicals must have exactly the same stuff inside them. You can subtract similar radicals by subtracting the coefficients. You need to simplify removing a perfect square factor or you'll just be done.

For example, we can subtract $5\sqrt{11}-2\sqrt{11}$. Both terms have an $11$ in their radical. To subtract, just subtract their coefficients.

$$\begin{align} 5\sqrt{11}-2\sqrt{11}&=(5-2)\sqrt{11}\\ &=3\sqrt{11} \end{align}$$

On the other hand, we cannot subtract $8\sqrt{6}-4\sqrt{3}$. The terms have different numbers in their radical. We would just leave the expression as is.

$$\begin{align} \sqrt{18} - \sqrt{200}&=\sqrt{9\times2} - \sqrt{100\times2}\\ &=3\sqrt{2} - 10\sqrt{2}\\ &=-7\sqrt{2} \end{align}$$

Great job! We started by breaking up the radicand into its prime factors, so that we can remove values to be coefficients. Then, we used our earlier skills to subtract radical expressions.