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How to simplify expressions and complex fractions

How to simplify the following kinds of expression: like terms, brackets, exponents and complex fractions (a fraction inside a fraction), each with checked examples.

1 منٹ کا مطالعہتازہ کاری 2 نومبر، 2026از Sumzo teamجائزہ: Sumzo team

To simplify an expression is to write it in a shorter, equivalent form — its value doesn’t change for any xx. The tools are always the same: expand brackets, collect like terms, apply the exponent rules, and cancel common factors. This lesson works through each, then shows how to simplify complex fractions, where a fraction sits inside another fraction.

Simplify the following expression: like terms

حل شدہ مثال

Easy: expand and collect

Simplify 4(2x−3)−5x+74(2x - 3) - 5x + 7.

حل دکھائیں
  1. Expand the bracket.
    8x−12−5x+78x - 12 - 5x + 7
  2. Collect the xx terms and the numbers.
    3x−53x - 5

جواب: 3x−53x - 5

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حل شدہ مثال

Medium: two brackets

Simplify (x+3)(x−2)−x(x+4)(x + 3)(x - 2) - x(x + 4).

حل دکھائیں
  1. Expand each product.
    x2+x−6−x2−4xx^2 + x - 6 - x^2 - 4x
  2. Collect like terms; the x2x^2 terms cancel.
    −3x−6-3x - 6

جواب: −3x−6-3x - 6

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Simplifying with exponent rules

  • xa⋅xb=xa+bx^a \cdot x^b = x^{a + b} — add exponents when multiplying.
  • xaxb=xa−b\frac{x^a}{x^b} = x^{a - b} — subtract when dividing.
  • (xa)b=xab(x^a)^b = x^{ab} — multiply for a power of a power.
  • x−n=1xnx^{-n} = \frac{1}{x^n} and x0=1x^0 = 1 (for x≠0x \neq 0).

For example, 6x5⋅2x24x3=12x74x3=3x4\frac{6x^5 \cdot 2x^2}{4x^3} = \frac{12x^7}{4x^3} = 3x^4.

How to simplify complex fractions

A complex fraction has a fraction in its numerator, its denominator, or both. The quickest method is to multiply the top and the bottom by the lowest common denominator of all the small fractions — that clears them in one step.

حل شدہ مثال

Medium: a numerical complex fraction

Simplify 12+1356\dfrac{\frac{1}{2} + \frac{1}{3}}{\frac{5}{6}}.

حل دکھائیں
  1. The small denominators are 2, 3 and 6; multiply top and bottom by 6.
    3+25\frac{3 + 2}{5}
  2. Simplify.
    55=1\frac{5}{5} = 1

جواب: 11

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حل شدہ مثال

Hard: a complex fraction with xx

Simplify 1+1x1−1x\dfrac{1 + \frac{1}{x}}{1 - \frac{1}{x}}.

حل دکھائیں
  1. Multiply the top and the bottom by xx.
    x+1x−1\frac{x + 1}{x - 1}
  2. Nothing cancels. Note the restrictions: x≠0x \neq 0 (from the small fractions) and x≠1x \neq 1.
    x+1x−1,x≠0,1\frac{x + 1}{x - 1}, \quad x \neq 0, 1

جواب: x+1x−1\frac{x + 1}{x - 1}, with x≠0x \neq 0 and x≠1x \neq 1

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Common mistakes

  • Combining unlike terms: 3x+2x23x + 2x^2 is already simplified.
  • Forgetting to distribute a minus sign: −x(x+4)=−x2−4x-x(x + 4) = -x^2 - 4x.
  • Cancelling terms rather than factors: x+2x\frac{x + 2}{x} is not 22.
  • Multiplying only the top of a complex fraction by the common denominator.

اکثر پوچھے گئے سوالات

How do you simplify complex fractions?

Multiply the numerator and the denominator by the lowest common denominator of every small fraction inside, then simplify what is left. Or simplify the top and bottom separately and divide.

What does “simplify the following” mean in algebra?

Rewrite each expression in its simplest equivalent form: no brackets left to expand, like terms combined, and no common factor left in a fraction.

How do I know when an expression is fully simplified?

When there are no like terms left to collect, no brackets to expand (unless factored form is asked for), and no factor shared by a numerator and denominator.

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