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.

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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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.

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.

examples with solutions for order of operations: roots

Exercise #1

What is the answer to the following?

$3^2-3^3$

Video Solution

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 firstcalculate 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.

Answer

$-18$

Exercise #2

Sovle:

$3^2+3^3$

Video Solution

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 firstcalculate 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.

Answer

36

Exercise #3

Solve:

$5^2\cdot4+3^3$

Video Solution

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 firstcalculate 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.

Answer

127

Exercise #4

Calculate and indicate the answer:

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

Video Solution

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 firstcalculate 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.

Answer

1

Exercise #5

Solve:

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

Video Solution

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: