Order of Operations: Roots

Since this is not an operation that affects the rest of the operations of the exercise, we do not have to solve them from left to right as we do with the rest of the operations.

Let's look at the following example:

$5+{\sqrt{49}}+4^3+(10\cdot3):2=$

To solve it, we start by performing the operations inside the parentheses.

$5+{\sqrt{49}}+4^3+30:2=$

Next, we move on to roots and powers.

$5+7+64+30:2=$

In the next step, we perform the multiplications and divisions.

$5+7+64+15=$

Once solved, we move on to the addition and subtraction operations.

$5+7+64+15= 91$

Let's consider the following example:

$3+{\sqrt 81}+2^3+(3\cdot2):1=$

To solve it, we start by performing the operations inside the parentheses.

$3+{\sqrt 81}+2^3+6:1=$

Next, we move on to roots and powers.

$3+9+2^3+6:1=$,$3+9+8+6:1=$

In the next step, we perform the multiplications and divisions.

$3+8+8+6=$

Finally, we move on to the addition and subtraction operations.

$3+8+8+6=25$

Now we will do the same exercise, but with a small variation:

$3\cdot({\sqrt 81}+2^3)+3-2:1=$

Since the root and the power are inside parentheses, we first need to simplify them in order to remove the parentheses.

$3\cdot(9+8)+3-2:1=$,$3\cdot17+3-2:1=$

Next, in line with the order of operations, we can move on to the multiplications and divisions (remember, from left to right).
$51+3-2:1=$,$51+3-2=$

Now we can proceed on to the last operations: addition and subtraction.
$51+3-2=52$

Now let's try this exercise:

$\sqrt{9}+\sqrt{49}+\sqrt{121}\times13-(4^2+6^2)\text{ }=$

Here, we start by performing the operations inside the brackets, which in this case are powers.

$\sqrt{9}+\sqrt{49}+\sqrt{121}\times13-(16+36)\text{ }=$

$\sqrt{9}+\sqrt{49}+\sqrt{121}\times13-(52)\text{ }=$

$\sqrt{9}+\sqrt{49}+\sqrt{121}\times13-52\text{ }=$

Next, we move on to the multiplications and divisions (remember, from left to right).

$3+7+11\times13-52=$

Then the multiplications and divisions (from left to right).

$3+7+143-52=$

Finally, we add and subtract.
$3+7+143-52=101$

${\sqrt9}\cdot{\sqrt4}+9^2\cdot6=$

In this exercise we see that there are no parentheses. Therefore, we solve the roots and powers first in order from left to right.

$3\cdot2+81\cdot6=$

Now we continue on to multiplications and divisions (from left to right).

$6+486=$

Finally, we add and subtract.
$6+486=492$

$3^2-2+{\sqrt49}=$

In this exercise we see that there are no parentheses either, so again we solve the roots and powers in order from left to right.

$9-2+{7}=$

Finally, we add and subtract.
$9-2+{7}=14$

$\sqrt[3]{27}+\left(\sqrt{2}\right)^2+\frac{\sqrt{16}}{\sqrt[3]{8}}+\sqrt{9}\times\sqrt{4} =$

In order to perform the addition of roots, we start by calculating the cube root of 27, which is 3. When we square the square root of 2, the root and the power cancel each other out, leaving us with a result of 2.

$3+2+\frac{\sqrt{16}}{\sqrt[3]{8}}+\sqrt{9}\times\sqrt{4}=$

In order to perform root multiplications and divisions, we first obtain the result of each root.

$3+2+\frac{4}{2}+3\times2=$

Next, we perform the division and multiplication.

$3+2+2+6=$

Finally, we calculate the sum.

$13$

• ${\sqrt 16} \cdot{\sqrt 4}+4^2\cdot10=$
• $12^2-7+{\sqrt 36}=$
• $6+{\sqrt 64}-4=$
• $2\cdot({\sqrt 32}+9)=$
• $2\cdot(3^3+{\sqrt 144})=$
• $(4^2+3)\cdot{\sqrt 9}=$
• $18^2-(100+{\sqrt 9})=$
• $({\sqrt 16}-2^2+6):2^2=$

If you liked this article, you may also be interested in the following:

• The order of operations
• Order of basic operations: addition, subtraction, multiplication and division
• Order of basic operations (powers)
• The 0 and the 1 in the order of operations
• Neutral elements / Neutral elements

On the Tutorela blog, you can find a variety of useful articles about mathematics!

When we have combined operations where both divisions and roots appear, the root is solved first and then the division.

Roots and the powers share the same level of importance within the order of operations. As these operations neither affect each other nor the rest of the operations, it is not necessary to perform them from right to left.

When we have operations or exercises combined with different operations, we must solve them in the following order:

1. Operations within parentheses (The order of operations is maintained within these).
2. Roots and powers.
3. Multiplications and divisions (from left to right).
4. Additions and subtractions (from left to right).

Before solving the roots, we must solve the operations inside the parentheses. Once this has been done, we proceed with solving the roots and powers.