TheSimplify Calculator rewrites an expression into an equivalent form that is easier to read, evaluate, or use, while providing step-by-step explanations. Enter the problem with numbers, variables, fractions, exponents, radicals, parentheses, and other mathematical symbols.
How to Simplify Expressions
Simplifying means rewriting an expression without changing its value. The right method depends on the expression’s structure. A numerical expression may need order of operations, while an algebraic expression may need distribution, combining like terms, or fraction reduction.
Following the Order of Operations
Order of operations applies to numerical expressions and algebraic expressions. Work through parentheses first, then exponents, multiplication and division, and finally addition and subtraction.
\[ 8+3(4-2)^2 \]
\[ =8+3(2)^2 \]
\[ =8+3(4)=20 \]
Following this order preserves the meaning of the original expression. Changing the order can produce a different value.
Combining Like Terms
Like terms have the same variable part and the same exponents. You can add or subtract their coefficients.
\[ 3x+7x-4=10x-4 \]
The terms \(3x\) and \(7x\) are like terms. The terms \(3x\) and \(4x^2\) are not, because one has \(x\) and the other has \(x^2\).
Using the Distributive Property
The distributive property means multiplying the outside factor by every term inside parentheses.
\[ 4(x+3)=4x+12 \]
A negative factor must also multiply every term.
\[ -2(3x-5)=-6x+10 \]
Distribution often reveals like terms that can be combined afterward.
Reducing Fractions and Algebraic Fractions
To simplify a numerical fraction, divide the numerator and denominator by their greatest common factor.
\[ \frac{42}{56}=\frac{3}{4} \]
For algebraic fractions, factor the numerator and denominator before canceling common factors.
\[ \frac{x^2-9}{x-3} = \frac{(x-3)(x+3)}{x-3} = x+3,\qquad x\ne3 \]
The restriction \(x\ne3\) remains because \(x=3\) made the original denominator zero. Canceling a common factor changes the appearance of the expression, but it does not remove a value excluded by the original expression.
Only factors can cancel. In \(\frac{x+3}{x}\), the \(x\) is not a factor of the entire numerator, so nothing cancels.
Simplifying Complex Fractions
A complex fraction has a fraction in its numerator, denominator, or both. You can simplify smaller fractions first or multiply by a common denominator.
\[ \frac{\frac{3}{4}}{\frac{9}{10}} = \frac{3}{4}\cdot\frac{10}{9} = \frac{5}{6} \]
This removes the stacked fraction structure without changing the value.
Applying Exponent Rules
Use exponent rules only when the bases and operations meet the rule’s conditions.
\[ x^3\cdot x^5=x^8 \]
\[ \frac{x^7}{x^2}=x^5,\qquad x\ne0 \]
\[ (x^3)^2=x^6 \]
\[ (ab)^3=a^3b^3 \]
\[ \left(\frac{a}{b}\right)^2=\frac{a^2}{b^2},\qquad b\ne0 \]
A zero exponent gives \(x^0=1\) when \(x\ne0\). A negative exponent moves the factor across the fraction bar:
\[ x^{-3}=\frac{1}{x^3},\qquad x\ne0 \]
Do not add exponents when bases differ. For example, \(x^2y^2\) does not equal \(x^4\). Fractional exponents connect naturally to radicals, such as \(x^{1/2}=\sqrt{x}\) when real-number conditions allow it.
Removing Unnecessary Parentheses
Parentheses affect both grouping and signs. A plus sign outside parentheses leaves the terms unchanged:
\[ +(x-6)=x-6 \]
A minus sign changes the sign of every term inside:
\[ -(x-6)=-x+6 \]
Check signs before removing parentheses. Missing one sign can change the entire expression.
Simplifying Radicals
To simplify square roots, look for perfect-square factors inside the radical.
\[ \sqrt{72}=\sqrt{36\cdot2}=6\sqrt{2} \]
Like radicals can be combined:
\[ 3\sqrt{5}+2\sqrt{5}=5\sqrt{5} \]
A square root is simplified when no perfect-square factor remains inside. The same idea works with higher roots, using perfect cubes for cube roots and matching perfect powers for other roots.
When a radical remains in a denominator, rationalizing the denominator may produce a standard equivalent form:
\[ \frac{4}{\sqrt{3}}=\frac{4\sqrt{3}}{3} \]
Factoring Expressions When It Helps
Factoring is sometimes useful during simplification, especially when it exposes a common factor or makes an algebraic fraction reducible. It is not required for every expression.
For example, factoring \(x^2-9\) as \((x-3)(x+3)\) makes the common factor in an algebraic fraction visible. For help finding those factors, the Factoring Calculator may be useful. The path is assumed from the provided site structure and should be verified before publishing.
Expanding and Simplifying
Some expressions need to be expanded before like terms become visible.
\[ (x+4)(x+2) \]
\[ =x^2+2x+4x+8 \]
\[ =x^2+6x+8 \]
The first step expands the product. The second step combines the like terms \(2x\) and \(4x\).
Simplifying Completely
Simplifying completely often takes more than one rule. After each step, check whether like terms remain, fractions can reduce, exponents can simplify, radicals contain perfect-power factors, or parentheses are still needed.
Not every rule applies to every expression. Simplification must preserve the expression’s value and any restrictions that came from denominators, radicals, or other operations.
If an equation remains after simplification, finding the variable’s value is a separate task. The Solve for x Calculator can help with that next stage; this assumed internal path should be verified before publishing.
Recognizing an Expression That Is Already Simplified
An expression does not always need to change. For example, \(4x+3y\) may already be simplified because \(4x\) and \(3y\) are unlike terms, and no other operation applies.
“Simplest form” does not always mean there is one unique-looking arrangement. Equivalent expressions can look different while having the same value for every allowed variable value.
Understanding Simplification
Simplifying rewrites an expression into an equivalent form. Depending on the problem, it can involve combining like terms, distributing multiplication, reducing fractions, applying exponent rules, simplifying radicals, removing parentheses, or factoring when that creates a more useful form.
An expression is not an equation. For example, \(4x+3x-2\) simplifies to \(7x-2\), but there is no value of \(x\) to solve for. In contrast, \(4x+3x=21\) simplifies to \(7x=21\), which can then be solved.
Simplifying, evaluating, and solving are different actions. The expression \(3x+5\) may already be simplified, but if \(x=4\), evaluating substitutes that known value:
\[ 3(4)+5=17 \]
Simplification preserves meaning while changing form. Evaluation produces a number from known values. Solving an equation finds variable values that make the equation true.
Worked Example
Simplify:
\[ 3(2x-5)+4x+7-2(x-3) \]
Distribute the multiplication across both parentheses.
\[ 3(2x-5)+4x+7-2(x-3)=6x-15+4x+7-2x+6 \]
Each outside factor multiplies every term inside its parentheses.
Rewrite the expression without parentheses.
\[ 6x-15+4x+7-2x+6 \]
The distributed terms are now separate and ready to combine.
Combine the like terms containing \(x\).
\[ 6x+4x-2x=8x \]
All three terms have the same variable part, \(x\).
Combine the constant terms.
\[ -15+7+6=-2 \]
The number terms add to \(-2\).
State the fully simplified expression.
\[ 3(2x-5)+4x+7-2(x-3)=8x-2 \]
No like terms or unnecessary parentheses remain.