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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Test your knowledge

Question 1

Solve the following exercise and circle the correct answer:

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.

Test your knowledge

Question 1

Which of the following is equivalent to \( 100^0 \)?

Examples with solutions for Order of Operations: Roots

Exercise #1

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 #2

$3\times3+3^2=$

Video Solution

Step-by-Step Solution

Let's recall the order of operations:

Parentheses

Exponents and Roots

Multiplication and Division

Addition and Subtraction

There are no parentheses in this problem, so we'll start with exponents:

3*3+3Ā² =

3*3+9 =

Let's continue to the next step, multiplication operations:

3*3+9 =

9 + 9 =

Now we're left with just a simple addition problem:

9+9= 18

And that's the solution!

Answer

18

Exercise #3

$6+\sqrt{64}-4=$

Video Solution

Step-by-Step Solution

To solve the expression $6+\sqrt{64}-4=$, we need to follow the order of operations (PEMDAS/BODMAS):

P: Parentheses (or Brackets)

E: Exponents (or Orders, i.e., powers and roots, etc.)

MD: Multiplication and Division (left-to-right)

AS: Addition and Subtraction (left-to-right)

In this expression, we first need to evaluate the square root since it falls under the exponent category:

$\sqrt{64} = 8$

Next, we substitute the computed value back into the expression:

$6+8-4$

We then perform the addition and subtraction from left to right:

$6+8 = 14$

$14-4 = 10$

Thus, the final answer is:

$10$

Answer

10

Exercise #4

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 #5

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.