# Order of Operations: Roots

🏆Practice powers and roots

As we have learned in previous lessons, when working with combined operations the order of the basic operations must be followed in order to get the correct result. However, before performing these the parentheses and then the roots and powers must first be solved.

Roots are very important in mathematical calculations. They are present in a variety of exercises ranging from algebraic problems for solving a second degree equation using the general formula, to geometric problems like determining the length of the hypotenuse of a right-angled triangle. Therefore, it is fundamental that we learn how to solve combined operations where this operation appears.

When we have simplified the root and power operations, we can continue solving the exercise according to the order of the basic operations: multiplications and divisions first, followed by additions and subtractions.

Let's revisit the order of the operations:

## Test yourself on powers and roots!

Which of the following is equivalent to $$100^0$$?

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.

$5+7+64+15= 91$

## Order of Operations Examples

### Exercise 1

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.

$3+8+8+6=25$

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### Exercise 2

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.
$51+3-2=52$

### Exercise 3

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).

$3+7+143-52=$

$3+7+143-52=101$

Do you know what the answer is?

### Exercise 4

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

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).

$6+486=$

$6+486=492$

### Exercise 5

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

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.

$9-2+{7}=$

$9-2+{7}=14$

### Exercise 6

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

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.

$13$

## Order of Operations: Root Exercises

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

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Do you think you will be able to solve it?

## Review Questions

### Which is done first, division or root?

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

### Which is done first, the root or the power?

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.

### What is the correct order when performing mathematical operations?

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).

### How do you solve combined operations with roots?

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.

Do you know what the answer is?

## examples with solutions for order of operations: roots

### Exercise #1

What is the answer to the following?

$3^2-3^3$

### Step-by-Step Solution

Remember that according to the order of operations, exponents come before multiplication and division, which come before addition and subtraction (and parentheses always before everything),

So first calculate the values of the terms in the power and then subtract between the results:

$3^2-3^3 =9-27=-18$Therefore, the correct answer is option A.

$-18$

### Exercise #2

Sovle:

$3^2+3^3$

### Step-by-Step Solution

Remember that according to the order of operations, exponents precede multiplication and division, which precede addition and subtraction (and parentheses always precede everything).

So first calculate the values of the terms in the power and then subtract between the results:

$3^2+3^3 =9+27=36$Therefore, the correct answer is option B.

36

### Exercise #3

Solve:

$5^2\cdot4+3^3$

### Step-by-Step Solution

Remember that according to the order of arithmetic operations, exponents precede multiplication and division, which precede addition and subtraction (and parentheses always precede everything).

So first calculate the values of the terms with exponents and then subtract the results:

$5^2\cdot4+3^3 =25\cdot4+27=100+27=127$Therefore, the correct answer is option B.

127

### Exercise #4

$(5-2)^2-2^3$

### Step-by-Step Solution

Remember that according to the order of arithmetic operations, exponents precede multiplication and division, which precede addition and subtraction (and parentheses always precede everything).

So first calculate the values of the terms with exponents and then subtract the results:

$(5-2)^2-2^3 =3^2-2^3=9-8=1$Therefore, the correct answer is option C.

1

### Exercise #5

Solve:

$\sqrt{4}\cdot4^2-5^2\cdot\sqrt{1}$

### Step-by-Step Solution

We simplify each term according to the order from left to right:

$\sqrt{4}=2$

$4^2=4\times4=16$

$5^2=5\times5=25$

$\sqrt{1}=1$

Now we rearrange the exercise accordingly:

$2\times16-25\times1$

Since there are two multiplication operations in the exercise, according to the order of operations we start with them and then subtract.

We put the two multiplication exercises in parentheses to avoid confusion during the solution, and solve from left to right:

$(2\times16)-(25\times1)=32-25=7$