How to Rewrite the Expression as a Simplified Expression Containing One Term

To rewrite an expression as a simplified expression containing one term, you combine everything into a single monomial — one coefficient multiplying one product of variables. This usually means multiplying out parentheses, combining like terms, and applying exponent rules so that no plus or minus signs remain between separate pieces. For example, 3x(2x) + 4x² simplifies to 10x².

Combine Like Terms First

Like terms share exactly the same variable part, including exponents. Only their coefficients differ. Add or subtract those coefficients and keep the variable part unchanged.

Take 5x² + 3x² − 2x². All three terms have x², so add the coefficients: 5 + 3 − 2 = 6. The result is 6x², a single term. A quick check before you start: list the variable part of each term and group identical ones. Anything left ungrouped stays separate, which means the expression genuinely has more than one term and cannot be reduced further.

Watch the signs. Subtraction distributes across the terms that follow it. In 7y − (2y − 3y), the parentheses hold two like terms that combine to −y, so you get 7y − (−y) = 8y. Treating it as 7y − 2y − 3y would wrongly give 2y. Rewrite subtraction as adding the opposite when you’re unsure.

Multiply Out Parentheses Using Exponent Rules

When a coefficient sits outside a product, multiply it across and add exponents for identical bases.

For 4a² · 3a³: multiply coefficients 4 × 3 = 12, then add exponents 2 + 3 = 5, giving 12a⁵. One coefficient, one variable, one term.

A slightly bigger case: 2x(3x² + 5x) distributes to 6x³ + 10x². These terms don’t share the same exponent, so they can’t combine — the expression stays as two terms. It only collapses to one term if the exponents match after multiplying, as in 2x(3x² + 2x²), which becomes 2x · 5x² = 10x³.

Raising a product to a power distributes over both the coefficient and each variable. For (3x²)²: square the coefficient, 3² = 9, and multiply each exponent by 2 to get x⁴. The result is 9x⁴. This rule works only when the base is a single product — (x + 3)² does not simplify this way.

Divide Carefully to Reach a Single Term

Division follows the inverse rule: divide coefficients and subtract exponents. In (12x⁵) ÷ (4x²), divide 12 by 4 to get 3, then subtract exponents 5 − 2 = 3, giving 3x³.

When dividing a sum by a single term, split the fraction first: (8x⁴ + 4x²) ÷ 2x² equals 8x⁴/2x² + 4x²/2x², which is 4x² + 2. That’s two terms, because the exponents differ. To end with one term, the whole numerator must share the divisor’s variable power.

Here’s a short worked example that combines several steps:

Simplify 5x(2x − x) + (3x)² − 4x².

Inside the parentheses, 2x − x = x, so the first part is 5x · x = 5x². Then (3x)² = 9x². Now you have 5x² + 9x² − 4x², all like terms: 5 + 9 − 4 = 10. The final answer is 10x².

Handle Radicals and Fractional Exponents

Radicals follow the same “one term” logic, provided the radicands match. √3 + √3 = 2√3, but √3 + √5 stays separate because the numbers under the sign differ.

To express a radical as a single term with an exponent, convert it first. The cube root of x⁵ is x^(5/3), using the rule that the nth root of xᵐ is x^(m/n). Once converted, the usual exponent rules apply. So (∜x³)(∜x⁵) becomes x^(3/4) · x^(5/4) = x^(8/4) = x², a clean single term. Some square roots also simplify under multiplication: √8 · √2 = √16 = 4. That works because √a · √b = √(ab) when both radicands are nonnegative.

Occasionally you can turn a two-term radical expression into one term by rationalizing. The expression 2/√2 simplifies to √2: multiply numerator and denominator by √2 to get 2√2/2 = √2. The key is that the denominator cancels cleanly, leaving one monomial.

Use Factoring and Distribution to Merge Polynomials

Expanding is not the only route to one term — sometimes factoring gets you there faster. If you see x(3 + 4), the sum inside is 7, so the whole thing is 7x. Doing the addition before multiplying avoids a two-term intermediate step.

The same trick handles common factors. In 6x² + 9x², factoring out 3x² gives 3x²(2 + 3) = 3x²(5) = 15x². You could combine the like terms directly, but factoring proves the result is one term by shrinking the parentheses to a single number.

A useful contrast: x(x + 1) distributes to x² + x, which stays as two terms because 1 is added to a different power of x. But x(x + x) becomes x(2x) = 2x², one term, because the inside collapses first. Spotting when the parentheses reduce to a plain number is the whole skill.

Most special products give multiple terms. (a + b)² = a² + 2ab + b² has three, and (a + b)(a − b) = a² − b² has two. Don’t expect these identities to collapse; they expand. The real single-term power tools are the exponent laws, not the binomial identities.

Check Your Work Step by Step

Verification matters because a term that looks combinable often isn’t. Rewrite all terms with explicit coefficients, write each term’s variable part in a list, group entries that match exactly, then add or subtract coefficients within each group and count the groups that remain. If exactly one group survives, you have your monomial. If two or more remain, the expression is already as simplified as it can be.

Common errors to avoid: combining x² and x, since different exponents mean different variable parts; adding exponents when multiplying inside a sum, as in x² + x³, which does not equal x⁵; multiplying exponents when multiplying powers of one base, since 3x² · 4x³ gives 12x⁵, not 12x⁶; and forgetting that a coefficient applies to every factor in the parentheses.

Run each result back through the rules once. A quick substitution, such as setting x = 2 in the original and the simplified forms, confirms they agree. If the two values differ, an exponent or sign slipped somewhere in the process.

Key Takeaways

• A single-term result, called a monomial, has one coefficient and one product of variables.
• Combine like terms by adding or subtracting coefficients while leaving the variable parts untouched.
• Multiply coefficients and add exponents for identical bases; divide coefficients and subtract exponents.
• Distribute subtraction signs carefully — treating −(2y − 3y) as −2y − 3y is a frequent mistake.
• Check whether the variable parts truly match before combining; if they don’t, a one-term answer isn’t possible, and the simplified multi-term form is the correct final result.