COMBINED OPERATIONS (EXERCISES SOLVED) - MATHS - 2025

The combined operations are mathematical operations to be performed to determine a result. These are taught for the first time in elementary school, although they are usually used in later courses, being key to solving higher mathematical operations.

A mathematical expression with combined operations is an expression where different types of calculations must be carried out, following a certain order of hierarchy, until all the operations in question have been carried out.

Example of combined operations

In the previous image, you can see an expression where different types of basic mathematical operations appear, therefore, it is said that this expression contains combined operations. The basic operations carried out are the addition, subtraction, multiplication, division and / or enhancement of mainly whole numbers.

Expressions and hierarchies of combined operations

As already said before, an expression with combined operations is an expression where mathematical calculations such as an addition, a subtraction, a product, a division and / or the calculation of a power must be performed.

These operations may involve real numbers, but for ease of understanding, only whole numbers will be used in this article.

Two expressions with different combined operations are as follows:

5 + 7 × 8-3

(5 + 7) x (8-3).

The expressions above contain the same numbers and the same operations. However, if the calculations are performed, the results will be different. This is due to the parentheses in the second expression and the hierarchy with which the first must be resolved.

What is the hierarchy for solving expressions with combined operations?

When there are grouping symbols such as parentheses (), brackets, or braces {}, what is inside each pair of symbols must always be resolved first.

In the case that there are no grouping symbols, the hierarchy is as follows:

- first solve the powers (if any)

- then the products and / or divisions are resolved (if any)

- in last place the additions and / or subtractions are solved

Solved exercises

Here are some examples where you have to solve expressions that contain combined operations.

Exercise 1

Solve the two operations presented above: 5 + 7 × 8-3 and (5 + 7) x (8-3).

Solution

Since the first expression does not have grouping signs, the hierarchy described above must be followed, therefore, 5+ 7 × 8- 3 = 5 + 56-3 = 58.

On the other hand, the second expression has grouping signs, so that we must first solve what is inside said signs and therefore, (5 + 7) x (8-3) = (12) x (5) = 60.

As said before, the results are different.

Exercise 2

Solve the following expression with combined operations: 3² - 2³x2 + 4 × 3-8.

Solution

In the given expression, you can see two powers, two products, an addition and a subtraction. Following the hierarchy, you must first solve the powers, then the products, and finally the addition and subtraction. Therefore, the calculations are as follows:

9 - 8 × 2 + 4 × 3 - 8

9 - 16 +12 - 8

-3.

Exercise 3

Calculate the result of the following expression with combined operations: 14 ÷ 2 + 15 × 2 - 3³.

Solution

In the expression in this example, there is a power, a product, a division, an addition and a subtraction, and therefore the calculations proceed as follows:

14 ÷ 2 + 15 × 2 - 27

7 + 30 - 27

10

The result of the given expression is 10.

Exercise 4

What is the result of the following expression with combined operations: 1 + 6 × 3 - 46 ÷ 2 + 4² ÷ 2?

Solution

The previous expression, as you can see, contains addition, subtraction, multiplication, division and empowerment. Therefore, it must be solved step by step, respecting the order of the hierarchy. The calculations are as follows:

1 + 6 × 3 - 46 ÷ 2 + 4² ÷ 2

1 + 6 × 3 - 46 ÷ 2 + 16 ÷ 2

1 + 18 - 23 + 8

3

In conclusion, the result is 3.

References

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