Algebra Simplification Techniques

Simplification of Algebraic Expressions

• Combining like terms is possible when terms have the same variable(s) and coefficient, e.g., 2x + 3x = 5x
• To remove parentheses, distribute the number outside the parentheses to each term inside, e.g., 2(3x + 2) = 6x + 4
• Numerical terms can be combined, e.g., 2 + 3 = 5
• To simplify expressions with negative terms, subtract the term from both sides of the equation, e.g., -2x + 3 = 5 -> -2x = 2 -> x = -1

Simplification of Fractions

• To simplify fractions, cancel out common factors by dividing both the numerator and denominator by the greatest common divisor (GCD), e.g., 12/16 = 3/4
• Fractions with negative terms can be simplified by simplifying the numerator and denominator separately, e.g., -3/-4 = 3/4
• Mixed numbers can be simplified by converting them to improper fractions and then simplifying the fraction, e.g., 2 1/3 = 7/3 -> 7/3 = 7/3 (already simplified)

Simplification of Indices and Surds

• Indices can be simplified by combining like indices, e.g., 2^3 × 2^4 = 2^(3+4) = 2^7
• Surds can be simplified by expressing them in their simplest form, e.g., √(16) = √(4^2) = 4
• Rationalizing denominators involves multiplying the numerator and denominator by the conjugate of the denominator, e.g., 1/√2 = 1/√2 × √2/√2 = √2/2

Sentence Structure in the Present Tense

Simple Sentences

• A simple sentence consists of a subject and a verb in its base form (except for third person singular, which adds -s/-es).
• The verb agrees with the subject in number (singular or plural).
• Examples of simple sentences in the present tense:
  • I write.
  • You write.
  • He/She/It writes.
  • We write.
  • They write.

Compound Sentences

• A compound sentence joins two or more independent clauses using a conjunction.
• Each independent clause has a subject and a verb in the present tense.
• Examples of compound sentences in the present tense:
  • I write, and she types.
  • They are working, but he is not.

Complex Sentences

• A complex sentence has an independent clause and one or more dependent clauses.
• The dependent clause is introduced by a subordinating conjunction and has a subject and a verb in the present tense.
• Examples of complex sentences in the present tense:
  • I write because I want to express myself.
  • Since I am tired, I will go to bed.

Sentence Fragments

• Sentence fragments are incomplete sentences that lack a subject, verb, or both.
• Fragments can be corrected by adding the missing element(s) to form a complete sentence.
• Examples of sentence fragments and their corrections:
  • Writing → I am writing.
  • In the morning → I exercise in the morning.