Simplify Expression Calculator

Simplifying an algebraic expression means rewriting it in a shorter, cleaner, or more standard form without changing its value. The simplified form is easier to read, evaluate, and use in further calculations.

Common simplification operations include:

• Combining like terms: $3x + 5x = 8x$
• Canceling common factors: $\frac{x^2 - 9}{x + 3} = x - 3$ (for $x \neq -3$)
• Reducing exponents: $\frac{x^5}{x^2} = x^3$
• Expanding and collecting: $(x+1)^2 - x^2 = 2x + 1$

A simplified expression is equivalent to the original for all values in the domain. Note that "simplest form" can depend on context — sometimes factored form is simpler, sometimes expanded form is.

Simplification is a core algebra skill used in solving equations, evaluating limits, integrating functions, and communicating mathematical results clearly.

1. Combine Like Terms

Group terms with the same variable and exponent, then add their coefficients.

Example: $3x^2 + 2x - x^2 + 5x - 7 = 2x^2 + 7x - 7$

2. Apply Exponent Rules

Key rules:

• $x^a \cdot x^b = x^{a+b}$
• $\frac{x^a}{x^b} = x^{a-b}$
• $(x^a)^b = x^{ab}$

Example: $\frac{x^5 \cdot x^2}{x^4} = x^{5+2-4} = x^3$

3. Factor and Cancel

For rational expressions, factor numerator and denominator, then cancel common factors.

Example: $\frac{x^2 - 9}{x + 3} = \frac{(x+3)(x-3)}{x+3} = x - 3$ (for $x \neq -3$)

4. Expand Products

Use distribution or special formulas:

• $(a+b)^2 = a^2 + 2ab + b^2$
• $(a-b)(a+b) = a^2 - b^2$

Example: $(2x+3)^2 - 4x^2 = 4x^2 + 12x + 9 - 4x^2 = 12x + 9$

5. Rationalize Denominators

Eliminate radicals from denominators by multiplying by the conjugate:

$\frac{1}{\sqrt{x}+1} \cdot \frac{\sqrt{x}-1}{\sqrt{x}-1} = \frac{\sqrt{x}-1}{x-1}$

6. Simplify Complex Fractions

Multiply numerator and denominator by the LCD of all inner fractions.

• Canceling terms instead of factors: $\frac{x + 3}{x + 5} \neq \frac{3}{5}$. You can only cancel common factors of the entire numerator and denominator.
• Forgetting domain restrictions: When canceling $(x+3)$ from $\frac{(x+3)(x-3)}{x+3}$, note that $x \neq -3$ in the original expression.
• Incorrect exponent arithmetic: $x^2 \cdot x^3 = x^5$, not $x^6$. And $\frac{x^5}{x^2} = x^3$, not $x^{2.5}$.
• Distributing exponents over sums: $(x + y)^2 \neq x^2 + y^2$. The correct expansion is $x^2 + 2xy + y^2$.
• Stopping too early: Always check if the result can be simplified further (e.g., factor out a remaining GCF).