Simplify Calculator
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To simplify an expression, combine like terms, cancel common factors and apply identities until nothing more can be reduced. For example, \(\frac{x^{2} – 9}{x^{2} + 5x + 6} = \frac{(x-3)(x+3)}{(x+2)(x+3)} = \frac{x – 3}{x + 2}\) for x ≠ −3 and x ≠ −2.
\(\frac{x^{2} – 9}{x^{2} + 5 x + 6}\)Use ^ for powers, * or a space for multiplication, sqrt( ), sin( ), ln( ). Fractions like 3/4 + 5/6 work too.
| Excluded values | \(x \neq -3,\ -2\) |
Show steps
- Start with \(\frac{x^{2} – 9}{x^{2} + 5 x + 6}\)
- Factor the numerator: \(x^{2} – 9 = \left(x – 3\right)\left(x + 3\right)\)
- Factor the denominator: \(x^{2} + 5x + 6 = \left(x + 2\right)\left(x + 3\right)\)
- Cancel the common factor \(\left(x + 3\right)\) from the top and bottom.
- Result: \(\frac{x – 3}{x + 2}\)
How to use the simplify calculator
- Type the expression. Implicit multiplication is fine: 3x, 2(x + 1), x y.
- Press Simplify. The answer appears with the steps, plus factored and expanded forms when they differ.
- For rational expressions, the calculator lists the excluded values, which the original expression does not allow.
Every answer is checked by evaluating the original and simplified expressions at several random points.
What "simplified" means
- Polynomials: no parentheses left and like terms combined, written in descending powers: \(2(x+3) – 4(x-1) = -2x + 10\).
- Rational expressions: numerator and denominator have no common factor: \(\frac{x^{2}-4}{x-2} = x + 2\) for \(x \neq 2\).
- Radicals: no perfect-square factor under the root: \(\sqrt{72} = 6\sqrt{2}\).
- Exponents: each base appears once: \((2x^{3}y^{2})^{3} = 8x^{9}y^{6}\).
Simplifying rational expressions step by step
- Factor the numerator and the denominator completely.
- Note the values that make the original denominator zero; they stay excluded.
- Cancel factors that appear in both (factors, not terms).
- Multiply out what is left if you want expanded form.
Example: \(\frac{6x^{2}+11x+3}{4x^{2}-9} = \frac{(3x+1)(2x+3)}{(2x-3)(2x+3)} = \frac{3x+1}{2x-3}\), with \(x \neq \pm\frac{3}{2}\).
Rules used most often
| Rule | Example |
|---|---|
| Distributive property | \(a(b + c) = ab + ac\) |
| Product of powers | \(x^{m}x^{n} = x^{m+n}\) |
| Quotient of powers | \(\frac{x^{m}}{x^{n}} = x^{m-n}\) |
| Power of a product | \((xy)^{n} = x^{n}y^{n}\) |
| Difference of squares | \(a^{2} – b^{2} = (a-b)(a+b)\) |
| Pythagorean identity | \(\sin^{2}x + \cos^{2}x = 1\) |
Common mistakes
- Cancelling terms instead of factors: \(\frac{x+3}{x}\) is not \(3\).
- Dropping excluded values after cancelling.
- Adding exponents of different bases: \(x^{2}y^{3}\) cannot be combined.
- Distributing a minus sign to only the first term inside parentheses.
Simplifying expressions practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
Quiz complete
Frequently asked questions
How do I simplify an algebraic expression?
Remove parentheses with the distributive property, combine like terms, and for fractions factor the top and bottom and cancel common factors.
Why does the calculator list excluded values?
Cancelling a factor such as (x + 3) hides the fact that the original expression is undefined at x = −3. The simplified form equals the original only where the original is defined.
Can it simplify expressions with several variables?
Yes. For example (a² − b²)/(a − b) simplifies to a + b, and 3x + 2y − x + 4y to 2x + 6y.
Can it simplify trigonometric expressions?
It applies common identities such as sin²x + cos²x = 1. Very involved trig expressions may stay in an equivalent but longer form.
How do I know the answer is right?
The calculator evaluates the original and the simplified expression at several random points and only shows forms that match.
Sources
- OpenStax College Algebra 2e: Rational Expressions
- Paul’s Online Math Notes: Rational Expressions
- Paul’s Online Math Notes: Factoring Polynomials
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