Beginning in middle school, we increasingly useletters（a, b, x, y, etc.）instead of numbers when writing expressions. Letters can represent numbers that have not yet been determined or that change, allowing us to create formulas that apply to many situations. First, learn the “rules for writing algebraic expressions” carefully.
Omitting Multiplication（×）
In algebraic expressions, themultiplication sign × is omitted, and the following rules are used.
- In the product of a number and a letter, write thenumber before the letter。
Example：a×3 → 3a、x×(−5) → −5x - In the product of 1 or −1 and a letter,omit 1。
Example：1×a → a、(−1)×x → −x - Write products of letters inalphabetical order。
Example：b×a → ab、y×x×3 → 3xy - Write products of the same letter as apower。
Example：a×a → a²、x×x×x → x³ - For the product of a parenthesized expression and a number, write the number first.
Example：(a+b)×4 → 4(a+b)
Example ①
Rewrite the following expressions according to the rules for writing algebraic expressions.
(1) y × 6 (2) x × y × x × (−2) (3) (a − b) × 5 (4) (−1) × a × a × b
Answer：
(1) 6y … Put the number first
(2) −2x²y … Put the number first, x×x=x², alphabetical order
(3) 5(a−b) … Keep the parentheses, put the number first
(4) −a²b … Omit 1, a×a=a², alphabetical order
Rewrite the following expressions according to the rules for writing algebraic expressions.
(1) y × 6 (2) x × y × x × (−2) (3) (a − b) × 5 (4) (−1) × a × a × b
Answer：
(1) 6y … Put the number first
(2) −2x²y … Put the number first, x×x=x², alphabetical order
(3) 5(a−b) … Keep the parentheses, put the number first
(4) −a²b … Omit 1, a×a=a², alphabetical order
Omitting Division（÷）
Write division infraction form. a ÷ b ＝ a/b（the top of the fraction bar is the numerator, and the bottom is the denominator）.
- x ÷ 3 → x/3
- a ÷ (−4) → a/(−4) ＝ −a/4（the sign may be moved before the fraction）
- (a + b) ÷ 2 → (a+b)/2（keep the parentheses around the numerator）
Example ②
Rewrite the following without using × or ÷.
(1) a × 4 − b ÷ 3 (2) (x + 2y) ÷ (−5)
Answer：
(1) 4a − b/3 … Put the number first for multiplication; use a fraction for division
(2) −(x+2y)/5 … ÷(−5) means multiplying by 1/(−5). Move the sign before the fraction
Rewrite the following without using × or ÷.
(1) a × 4 − b ÷ 3 (2) (x + 2y) ÷ (−5)
Answer：
(1) 4a − b/3 … Put the number first for multiplication; use a fraction for division
(2) −(x+2y)/5 … ÷(−5) means multiplying by 1/(−5). Move the sign before the fraction
Key Points
- Omit ×：put numbers before letters, omit 1, and write letters in alphabetical order
- Write products of the same letter as powers（a×a×a = a³）
- Convert ÷ into a fraction：a÷b = a/b（a is the numerator and b is the denominator）
- Keep parenthesized expressions intact, and put a number or denominator before or after them
Practice Problems
- Write the following according to the rules for algebraic expressions.
(1) b × a × (−1) (2) x × x × y × y × y (3) (a + 3) × 2 - Write the following according to the rules for algebraic expressions.
(1) m ÷ 8 (2) (p − q) ÷ 3 (3) x × 5 ÷ y - Restore the following expression to a form using × and ÷：3a²b / 4
Answers and Explanations
- Answer：(1) −ab (2) x²y³ (3) 2(a+3)
Explanation：(1) −1×b×a → omit 1 and use alphabetical order to get −ab。(2) x×x=x²、y×y×y=y³ → x²y³。(3) Put the number first to get 2(a+3)。 - Answer：(1) m/8 (2) (p−q)/3 (3) 5x/y
Explanation：Convert ÷ into a fraction. (3) Change x×5 to 5x, then divide by y to get 5x/y。 - Answer：3 × a × a × b ÷ 4
Explanation：3a² is 3×a×a. Multiply this by b. ÷4 is the final denominator. Therefore, 3×a×a×b÷4。