Identity transformations of expressions

In this publication, we will consider the main types of identical transformations of algebraic expressions, accompanying them with formulas and examples to demonstrate their application in practice. The purpose of such transformations is to replace the original expression with an identically equal one.

Rearranging terms and factors

In any sum, you can rearrange the terms.

a + b = b + a

In any product, you can rearrange the factors.

a ⋅ b = b ⋅ a

examples:

• 1 + 2 = 2 + 1
• 128 ⋅ 32 = 32 ⋅ 128

Grouping terms (multipliers)

If there are more than 2 terms in the sum, they can be grouped by parentheses. If required, you can first swap them.

a + b + c + d = (a + c) + (b + d)

In the product, you can also group the factors.

a ⋅ b ⋅ c ⋅ d = (a ⋅ d) ⋅ (b ⋅ c)

examples:

• 15 + 6 + 5 + 4 = (15 + 5) + (6 + 4)
• 6 ⋅ 8 ⋅ 11 ⋅ 4 = (6 ⋅ 4 ⋅ 8) ⋅ 11

Addition, subtraction, multiplication or division by the same number

If the same number is added or subtracted to both parts of the identity, then it remains true.

If a + b = c + dthen (a + b) ± e = (c + d) ± e.

Also, equality will not be violated if both of its parts are multiplied or divided by the same number.

If a + b = c + dthen (a + b) ⋅/: e = (c + d) ⋅/: e.

examples:

• 35 + 10 = 9 + 16 + 20 ⇒ (35 + 10) + 4 = (9 + 16 + 20) + 4
• 42 + 14 = 7 ⋅ 8 ⇒ (42 + 14) ⋅ 12 = (7 ⋅ 8) ⋅ 12

Replacing a Difference with a Sum (often a Product)

Any difference can be represented as a sum of terms.

a – b = a + (-b)

The same trick can be applied to division, i.e. replace frequent with product.

a : b = a ⋅ b^-1

examples:

• 76 – 15 – 29 = 76 + (-15) + (-29)
• 42 : 3 = 42 ⋅ 3^-1

Performing arithmetic operations

You can simplify a mathematical expression (sometimes significantly) by performing arithmetic operations (addition, subtraction, multiplication and division), taking into account the generally accepted order of execution:

• first we raise to a power, extract the roots, calculate logarithms, trigonometric and other functions;
• then we perform the actions in brackets;
• lastly – from left to right, perform the remaining actions. Multiplication and division take precedence over addition and subtraction. This also applies to expressions in parentheses.

examples:

• 14 + 6 ⋅ (35 – 16 ⋅ 2) + 11 ⋅ 3 = 14 + 18 + 33 = 65
• 20 : 4 + 2 ⋅ (25 ⋅ 3 – 15) – 9 + 2 ⋅ 8 = 5 + 120 – 9 + 16 = 132

Bracket expansion

Parentheses in an arithmetic expression can be removed. This action is performed according to certain ones – depending on which signs (“plus”, “minus”, “multiply” or “divide”) are before or after the brackets.

examples:

• 117 + (90 – 74 – 38) = 117 + 90 – 74 – 38
• 1040 – (-218 – 409 + 192) = 1040 + 218 + 409 – 192
• 22⋅(8+14) = 22 ⋅ 8 + 22 ⋅ 14
• 18 : (4 – 6) = 18: 4-18: 6

Bracketing the Common Factor

If all the terms in the expression have a common factor, it can be taken out of brackets, in which the terms divided by this factor will remain. This technique also applies to literal variables.

examples:

• 3 ⋅ 5 + 5 ⋅ 6 = 5⋅(3+6)
• 28 + 56 – 77 = 7 ⋅ (4 + 8 – 11)
• 31x + 50x = x ⋅ (31 + 50)

Application of abbreviated multiplication formulas

You can also use to perform identical transformations of algebraic expressions.

examples:

• (31 + 4)² = 31² + 2 ⋅ 31 ⋅ 4 + 4² = 1225
• 26² – 7² = (26 – 7) ⋅ (26 + 7) = 627