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+49ā+43+(10ā 3):2=
To solve it, we start by performing the operations inside the parentheses.
5+49ā+43+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+8ā1+23+(3ā 2):1=
To solve it, we start by performing the operations inside the parentheses.
3+8ā1+23+6:1=
Next, we move on to roots and powers.
3+9+23+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.
3+2+38ā16āā+9āĆ4ā=
In order to perform root multiplications and divisions, we first obtain the result of each root.
3+2+24ā+3Ć2=
Next, we perform the division and multiplication.
3+2+2+6=
Finally, we calculate the sum.
13
Order of Operations: Root Exercises
1ā6ā 4ā+42ā 10=
122ā7+3ā6=
6+6ā4ā4=
2ā (3ā2+9)=
2ā (33+1ā44)=
(42+3)ā 9ā=
182ā(100+9ā)=
(1ā6ā22+6):22=
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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?
32ā33
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:
32ā33=9ā27=ā18Therefore, the correct answer is option A.
Answer
ā18
Exercise #2
Sovle:
32+33
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:
32+33=9+27=36Therefore, the correct answer is option B.
Answer
36
Exercise #3
Solve:
52ā 4+33
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:
52ā 4+33=25ā 4+27=100+27=127Therefore, the correct answer is option B.
Answer
127
Exercise #4
Solve:
4āā 42ā52ā 1ā
Video Solution
Step-by-Step Solution
We simplify each term according to the order from left to right:
4ā=2
42=4Ć4=16
52=5Ć5=25
1ā=1
Now we rearrange the exercise accordingly:
2Ć16ā25Ć1
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:
(2Ć16)ā(25Ć1)=32ā25=7
Answer
7
Exercise #5
Calculate and indicate the answer:
(5ā2)2ā23
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:
(5ā2)2ā23=32ā23=9ā8=1Therefore, the correct answer is option C.