How to Simplify Algebraic Expressions Step by Step

What simplify actually means

An expression such as $3(2x - 5) - 2(x - 4)$ has no equals sign, so there is nothing to solve. Simplifying it means finding an equivalent expression, one that gives the same value for every allowed $x$, that is shorter and easier to use. Here that is $4x - 7$.

Equivalent is the important word. Every step must be a valid algebraic rule, not a guess that happens to look neater. A useful test is to substitute a value: at $x = 2$ the original gives $3(-1) - 2(-2) = -3 + 4 = 1$, and $4(2) - 7 = 1$ as well. Matching at one point does not prove two expressions are equivalent, but a mismatch proves an error.

Combining like terms

Like terms have exactly the same variables raised to exactly the same powers; only their coefficients may differ. $3x^2$ and $-7x^2$ are like terms. $3x^2$ and $3x$ are not, and neither are $xy$ and $x^2y$.

To combine like terms, add their coefficients and keep the variable part unchanged:

Underline or colour each group of like terms before combining, and carry each term's sign with it. In $4x + 3y - x$, the term is $-x$, not $x$.

The distributive property and expanding brackets

The distributive property $a(b + c) = ab + ac$ removes brackets. The factor outside multiplies every term inside, and a minus sign in front of a bracket is a factor of $-1$.

• $-2(x - 4) = -2x + 8$, because $-2 \times -4 = +8$.
• $-(3x - y + 1) = -3x + y - 1$.
• $(x + 2)(x - 5) = x^2 - 5x + 2x - 10 = x^2 - 3x - 10$: multiply each term of the first bracket by each term of the second.
• $(2x - 3)^2 = (2x - 3)(2x - 3) = 4x^2 - 12x + 9$, not $4x^2 + 9$.

Squaring a sum term by term, as in $(a + b)^2 = a^2 + b^2$, is one of the most frequent algebra errors. The correct identity is $(a + b)^2 = a^2 + 2ab + b^2$.

Exponent rules

Products and quotients of powers simplify with a short set of rules, valid for any non-zero base:

The exponent calculator applies these rules one at a time if you want to check a longer expression.

Simplifying algebraic fractions

A rational expression is simplified by factoring the numerator and the denominator completely, then cancelling factors that appear in both. You may only cancel factors, never terms that are added.

The restriction $x \neq -3$ must be kept even though $(x + 3)$ no longer appears, because the original expression is undefined there. To add fractions, use a common denominator first:

If factoring is the sticking point, the guide to factoring polynomials covers every method used here, and the rational expression calculator shows the cancellation step by step.

Simplifying square roots

Take out perfect-square factors using $\sqrt{ab} = \sqrt{a}\sqrt{b}$ for non-negative $a$ and $b$. Then combine like radicals the same way as like terms.

• $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$ and $\sqrt{18} = 3\sqrt{2}$, so $\sqrt{50} + \sqrt{18} = 8\sqrt{2}$.
• $\sqrt{12} \cdot \sqrt{3} = \sqrt{36} = 6$.
• $\sqrt{9 + 16} = \sqrt{25} = 5$, which is not $\sqrt{9} + \sqrt{16} = 7$. Roots do not split over addition.

A sensible order of operations

1. Expand brackets with the distributive property, taking care with negative signs.
2. Apply exponent rules to products, quotients and powers.
3. Combine like terms.
4. For fractions, factor completely and cancel common factors, keeping the restrictions.
5. Write the result in a standard order, usually by descending powers: $3x^2 - x + 4$.
6. Spot-check by substituting a simple value into the original and the result.

Mistakes that change the value

• Cancelling terms instead of factors: $\frac{x + 6}{x + 3}$ is not $2$ and not $\frac{6}{3}$.
• Distributing a minus sign to the first term only.
• Adding exponents when multiplying different bases: $x^2 y^3$ cannot be combined into one power.
• Writing $(a + b)^2 = a^2 + b^2$ or $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$.
• Dropping the domain restrictions after cancelling.

Paste any expression into the simplify calculator to compare its steps with yours. Once an expression is simple, solving the related equation is much easier; see how to solve linear equations.

A longer worked example

Real exercises usually combine several of the rules above. Working through one carefully shows how the order of steps keeps the expression manageable.

Notice that the subtraction in front of $(x + 1)^2$ was handled only after the square had been expanded inside brackets. Expanding first and then distributing the minus sign to every term is what prevents the classic sign error.

What counts as fully simplified?

Teachers and textbooks use a few shared conventions for a final answer. An expression is usually considered fully simplified when there are no brackets left that could be expanded, unless a factored form was asked for; no two terms are alike; no exponent is negative or zero; no fraction has a common factor in its numerator and denominator; no square root contains a perfect-square factor; and no denominator contains a square root.

The last rule is handled by rationalising. To rationalise $\frac{6}{\sqrt{3}}$, multiply the top and bottom by $\sqrt{3}$ to get $\frac{6\sqrt{3}}{3} = 2\sqrt{3}$. Both forms have the same value, about $3.464$, but the second is the conventional final form.

Sometimes the simplest form depends on the purpose. The expanded form $x^2 - 3x - 10$ is convenient for reading off coefficients, while the factored form $(x + 2)(x - 5)$ shows the zeros immediately. If an exercise does not say which form it wants, either is equivalent, but most teachers expect the expanded form for sums and the factored, cancelled form for fractions.

If two different-looking answers both survive the substitution check at two or three values, they are almost certainly the same expression written two ways.

Simplifying with more than one variable

Expressions with several variables follow exactly the same rules, but it is easier to lose track of terms. Sorting the terms by their variable part before combining helps. In $3a^2b - 2ab^2 + 5ab - a^2b + 4ab^2$, the like groups are $a^2b$ terms, $ab^2$ terms and $ab$ terms. Combining each group gives $2a^2b + 2ab^2 + 5ab$. Nothing further combines, because each remaining term has a different variable part, but every term has a common factor of $ab$, so the expression can also be written as $ab(2a + 2b + 5)$.

Writing variables in alphabetical order within each term, for example $ba$ as $ab$, makes like terms easier to spot. It is also a common convention in final answers.

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