Chapter № 𝐼; Definitions, Substitutions
ALGEBRA
CHAPTER 𝐼.
Definitions. Substitutions.
𝟏. Algebra treats of quantities as in Arithmetic, but with greater generality; for while the quantities used in arithmetical processes are denoted by ƒigures, which have a single definite value, algebraic quantities are denoted by symbols, which may have any value we choose to assign to them.
The symbols of quantity employed are usually the letters of our own alphabet; and, though there is no restriction as to the numerical values a symbol may represent, it is understood that in the same piece of work it keeps the same value throughout. Thus, when we say "let 𝑎 equal 𝟏," we do not mean that 𝑎 must have the value 𝟏 always, but only in the particular example we are considering. Moreover, we may operate with symbols without assigning to them any particular numerical value; indeed it is with such operations that Algebra is chiefly concerned. We begin with the definitions of Algebra, premising that the symbols +, -, ×, ÷, (), = will have the same meanings as in Arithmetic. Also, for the present, it will be assumed that all the algebraic symbols employed represent integral numbers.
𝟐. An algebraic expression is a collection of symbols; it may consist of one or more terms, which are the parts separated from each other by the signs + and —. Thus, 𝟕𝑎 + 𝟓𝑏 — 𝟑𝑐 — 𝒙 + 𝟐𝒚 is an expression consisting of five terms.
Note. When no sign precedes a term the sign + is understood.
𝟑. Expressions are either simple or compound. A simple expression consists of one term, as 𝟓𝑎. A compound expression consists of two or more terms. Compound expressions may be further distinguished. Thus an expression of two terms, as 𝟑𝑎 — 𝟐𝑏, is often called a binomial, and one of three terms, as 𝟐𝑎 + 𝟑𝑏 + 𝑐, a trinomial. Simple expressions are frequently spoken of as monomials, and compound expressions as multinomials or polynomials.
𝟒. When two or more quantities are multiplied together the result is called the product. One important difference between the notation of Arithmetic and Algebra should be here remarked. In Arithmetic the product of 𝟐 and 𝟑 is written 𝟐 × 𝟑, whereas in Algebra the product of 𝑎 and b may be written in any of the forms 𝑎 × 𝑏, 𝑎 · 𝑏, or 𝑎𝑏. The form 𝑎𝑏 is the most usual. Thus, if 𝑎 = 𝟐, 𝑏 = 𝟑, the product 𝑎𝑏 = 𝑎 × 𝑏 = 𝟐 × 𝟑 = 𝟔; but in Arithmetic 𝟐𝟑 means "twenty-three," or 𝟐 × 𝟏𝟎 + 𝟑.
𝟓. Each of the quantities multiplied together to form a product is called a factor of the product. Thus 𝟓, 𝑎, 𝑏 are the factors of the product 𝟓𝑎𝑏.
Note. The beginner should carefully notice the difference between term and factor.
𝟔. When one of the factors of an expression is a numerical quantity, it is called the coefficient of the remaining factors. Thus, in the expression 𝟓 𝑎𝑏, 𝟓 is the coefficient. But the word coefficient is also used in a wider sense, and it is sometimes convenient to consider any factor, or factors, of a product as the coefficient of the remaining factors. Thus, in the product 𝟔 𝑎𝑏𝑐, 𝟔 𝑎 may be appropriately called the coefficient of 𝑏𝑐. A coefficient which is not merely numerical is sometimes called a literal coefficient.
Note. When the coefficient is unity it is usually omitted, and we write simply 𝑎, instead of 𝟏 𝑎.
𝟕. A power of a quantity is the product obtained by repeating that quantity any number of times as a factor, and is expressed by writing the number of factors to the right of the quantity and above it. Thus,
𝑎 × 𝑎 is called the second power of 𝑎, and is written a²;
𝑎 × 𝑎 × 𝑎 is called the thirg power of 𝑎, and is written a³;
and so on.
The index or exponent is the number which expresses the power of any quantity. Thus 𝟐, 𝟓, 𝟕 are respectively the indices of 𝑎², 𝑎⁵, 𝑎⁷.
Note. 𝑎² is usually read "𝑎 squared"; 𝑎³ is read "𝑎 cubed"; 𝑎⁴ is read "𝑎 to the fourth"; and so on. When the index is unity it is omitted, and we write simply 𝑎, instead of 𝑎¹. Thus 𝑎, 𝟏 𝑎, 𝑎¹, and 𝟏 𝑎¹ all have the same meaning.
The expression 𝟎⁰ is read as "zero to the power of zero" and is generally considered undefined.
𝟖. The beginner must be careful to distinguish between coefficient and index.
Ex. 𝟏. What is the difference in meaning between 𝟑𝑎 and 𝑎³ ?
By 𝟑𝑎 we mean the product of the quantities 𝟑 and 𝑎.
By 𝑎³ we mean the product of the quantities 𝑎, 𝑎, 𝑎.
Thus, if 𝑎 = 𝟒,
𝟑𝑎 = 𝟑 × 𝑎 = 𝟑 × 𝟒 = 𝟏𝟐;
𝑎³ = 𝑎 × 𝑎 × 𝑎 = 𝟒 × 𝟒 × 𝟒 = 𝟔𝟒. Ex. 𝟐. If 𝑏 = 𝟓, distinguish between 𝟒𝑏² and 𝟐𝑏⁴.
Here 𝟒𝑏² = 𝟒 × 𝑏 × 𝑏 = 𝟒 × 𝟓 × 𝟓 = 𝟏𝟎𝟎;
Ex. 𝟑. If 𝒙 = 𝟏, find the value of 𝟓𝒙⁴.
Here 𝟓𝒙⁴ = 𝟓 × 𝒙 × 𝒙 × 𝒙 × 𝒙 = 𝟓 × 𝟏 × 𝟏 × 𝟏 × 𝟏 = 𝟓.
Note. The beginner should observe that every power of 𝟏 is 𝟏.
Ex. 𝟒. If 𝑎 = 𝟒, 𝒙 = 𝟏, find the value of 𝟓𝒙ᵃ.
𝟓𝒙ᵃ = 𝟓 × 𝒙ᵃ = 𝟓 × 𝟏⁴ = 𝟓 × 𝟏 = 𝟓.
𝟗. The Sign of Continuation, ... , is read "and so on."
𝟏𝟎. The Sign of Deduction, ∴, is read "therefore" or "hence."
And The Sign of Induction, ∵, is read "because" or "since."
𝟏𝟏. In arithmetical multiplication the order in which the factors of a product are written is immaterial. Thus 𝟑 × 𝟒 = 𝟒 × 𝟑. In like manner in Algebra 𝑎𝑏 and 𝑏𝑎 each denote the product of the two quantities represented by the letters 𝑎 and 𝑏, and have therefore the same value. Although it is immaterial in what order the factors of a product are written, it is usual to arrange them alphabetically. Fractional coefficients which are greater than unity are usually kept in the form of improper fractions. Ex. If 𝑎 = 𝟔, 𝒙 = 𝟕, 𝒛 = 𝟓, find the value of 𝟏𝟑/𝟏𝟎𝑎𝒙𝒛. Here 𝟏𝟑/𝟏𝟎𝑎𝒙𝒛 = 𝟏𝟑/𝟏𝟎 × 𝟔 × 𝟕 × 𝟓 = 𝟐𝟕𝟑.
EXAMPLES I. 𝑎.
If 𝑎 = 𝟕, 𝑏 = 𝟐, 𝑐 = 𝟏, 𝒙 = 𝟓, 𝒚 = 𝟑, find the value of
𝟏. 𝟏𝟒𝒙.
𝟐. 𝒙³.
𝟑. 𝟑𝑎𝒙.
𝟒. 𝑎³.
𝟓. 𝟓𝑏𝒚.
𝟔. 𝑏⁵.
𝟕. 𝟑𝑏².
𝟖. 𝟐𝑎𝒙.
𝟗. 𝟔𝑐⁴.
𝟏𝟎. 𝟒𝒚³.
If 𝑎 = 𝟖, 𝑏 = 𝟓, 𝑐 = 𝟒, 𝒙 = 𝟏, 𝒚 = 𝟑, find the value of
𝟏𝟏. 𝟑𝑐².
𝟏𝟐. 𝟕𝒚³.
𝟏𝟑. 𝟓𝑎𝑏.
𝟏𝟒. 𝟗𝒙𝒚.
𝟏𝟓. 𝟖𝑏³.
𝟏𝟔. 𝟑𝒙⁵.
𝟏𝟕. 𝒙⁸.
𝟏𝟖. 𝟕𝒚⁴.
𝟏𝟗. 𝑐ˣ.
𝟐𝟎. 𝑏ʸ.
𝟐𝟏. 𝒚ᶜ.
𝟐𝟐. 𝒙ᵇ.
𝟐𝟑. 𝒚ᵇ.
𝟐𝟒. 𝑎ʸ.
𝟐𝟓. 𝑏ˣ.
If 𝑎 = 𝟓, 𝑏 = 𝟏, 𝑐 = 𝟔, 𝒙 = 𝟒, find the value of
𝟐𝟔. 𝟏/𝟒 𝒙⁴.
𝟐𝟕. 𝟓/𝟏𝟐 𝑐³.
𝟐𝟖. 𝟑/𝟖 𝒙³.
𝟐𝟗. 𝟏/𝟏𝟎 𝑎𝒙.
𝟑𝟎. 𝟑ˣ.
𝟑𝟏. 𝟐ᶜ.
𝟑𝟐. 𝟖ᵇ.
𝟑𝟑. 𝟕ˣ.
𝟑𝟒. 𝟕/𝟏𝟓 𝑎𝑐𝒙.
𝟑𝟓. 𝟏/𝟖 𝑏𝑐𝒙.
𝟏𝟐. When several different quantities are multiplied together a notation similar to that of Art. 𝟕 is adopted. Thus 𝑎𝑎𝑏𝑏𝑏𝑏𝑐𝑑𝑑𝑑 is written 𝑎²𝑏⁴𝑐𝑑³. And conversely 𝟕𝑎³𝑐𝑑² has the same meaning as 𝟕 × 𝑎 × 𝑎 × 𝑎 × 𝑐 × 𝑑 × 𝑑.
Ex. 𝟏. If 𝒙 = 𝟓, 𝒚 = 𝟑, find the value of 𝟒𝒙²𝒚³.
𝟒𝒙²𝒚³ = 𝟒 × 𝟓² × 𝟑³ = 𝟒 × 𝟐𝟓 × 𝟐𝟕 = 𝟐𝟕𝟎𝟎.
Ex. 𝟐. If 𝑎 = 𝟒, 𝑏 = 𝟗, 𝒙 = 𝟔, find the value of 𝟖𝑏𝒙²/𝟐𝟕𝑎³.
𝟖𝑏𝒙²/𝟐𝟕𝑎³ = 𝟖 × 𝟗 × 𝟔²/𝟐𝟕 × 𝟒³ = 𝟖 × 𝟗 × 𝟑𝟔/𝟐𝟕 × 𝟔𝟒 = 𝟐𝟓𝟗𝟐/𝟏𝟕𝟐𝟖 = 𝟑/𝟐 = 𝟏 𝟏/𝟐.
𝟏𝟑. If one factor of a product is equal to 𝟎, the product must be equal to 𝟎, whatever values the other factors may have. A factor is usually called a zero factor.
For instance, if 𝒙 = 𝟎, then 𝑎𝑏³𝒙𝒚² contains a zero factor. Therefore 𝑎𝑏³𝒙𝒚² = when 𝒙 = 𝟎, whatever be the values of 𝑎, 𝑏, 𝒚.
Again, if 𝑐 = 𝟎, then 𝑐³ = 𝟎; therefore 𝑎𝑏²𝑐³ = 𝟎, whatever values 𝑎 and 𝑏 may have.
Note. Every power of 𝟎 is 𝟎. And 𝑎⁰ is always read as 1, provided that 𝑎 ≠ 𝟎.
EXAMPLES I. 𝑏.
If 𝑎 = 𝟕, 𝑏 = 𝟐, 𝑐 = 𝟎, 𝒙 = 𝟓, 𝒚 = 𝟑 find the value of
𝟏. 𝟒𝑎𝒙².
𝟐. 𝑎³𝑏.
𝟑. 𝟖𝑏²𝒚.
𝟒. 𝟑𝒙𝒚².
𝟓. 𝟑/𝟒𝑏²𝒙.
𝟔. 𝟓/𝟔 𝑏³𝒚².
𝟕. 𝟐/𝟓 𝒙𝒚⁴.
𝟖. 𝑎³𝑐 .
𝟗. 𝑎²𝑐𝒚.
𝟏𝟎. 𝟖𝒙³𝒚.
If 𝑎 = 𝟐, 𝑏 = 𝟑, 𝑐 = 𝟏, 𝒑 = 𝟎, 𝒕 = 𝟒, 𝒓 = 𝟔 find the value of
𝟏𝟏. 𝟑𝑎²𝒓.
𝟏𝟐. 𝟖𝑎𝑏².
𝟏𝟑. 𝟔𝑎³𝑐/𝑏².
𝟏𝟒. 𝟒𝑐𝒓²/𝟗𝑎³.
𝟏𝟓. 𝟑𝑎²𝑏𝑐.
𝟏𝟔. 𝟓/𝟔 𝑏𝑎ʳ.
𝟏𝟕. 𝟖𝑏/𝟗𝑎ʳ.
𝟏𝟖. 𝟓𝑎ᵇ𝑐ʳ.
𝟏𝟗. 𝟐𝑎²𝒑/𝟕𝒓.
𝟐𝟎. 𝟑ᵃ𝟐ᵇ.
𝟐𝟏. 𝟐𝒓𝑎⁵.
𝟐𝟐. 𝑐ᵇ𝑏ᵗ.
𝟐𝟑. 𝟓𝑎ʳ𝑏ᵗ/𝟔𝟒𝒓ᵃ.
𝟐𝟒. 𝟐𝟕𝑎ᵗ/𝟑𝟐.
𝟐𝟓. 𝟔𝟒𝒕ʳ.
𝟏𝟒. Definition. The square root of any proposed expression is that quantity whose square, or second power, is equal to the given expression. Thus the square root of 𝟖𝟏 is 𝟗, because 𝟗² = 𝟖𝟏.
The square root of 𝑎 is denoted by ²√𝑎, or more simply √𝑎.
Similarly the cube, fourth, fifth, etc., root of any expression is that quantity whose third, fourth, fifth, etc., power is equal to the given expression.
The roots are denoted by the symbols ³√, ⁴√, ⁵√, etc.
Examples:
³√𝟐𝟕; because (∴) 𝟑³ = 𝟐𝟕
⁵√𝟑𝟐; because (∴) 𝟐⁵ = 𝟑𝟐
The symbol √ is sometimes called the radical sign.
Ex. 𝟏. Find the value of 𝟓√(𝟔 𝑎³𝑏⁴𝑐), when 𝑎 = 𝟑, 𝑏 = 𝟏, 𝑐 = 𝟖.
𝟓√(𝟔 𝑎³𝑏⁴𝑐) = 𝟓 × √(𝟔 × 𝟑³ × 𝟏⁴ × 𝟖)
= 𝟓 × √(𝟔 × 𝟐𝟕 × 𝟖) = 𝟓 × √𝟏𝟐𝟗𝟔 = 𝟓 × 𝟑𝟔 = 𝟏𝟖𝟎.
Ex. 𝟐. Find the value of ³√(𝑎𝑏⁴/𝟖𝒙³), when 𝑎 = 𝟗, 𝑏 = 𝟑, 𝒙 = 𝟓.
³√(𝑎𝑏⁴/𝟖𝒙³) = ³√(𝟗 × 𝟑⁴/𝟖 × 𝟓³) = ³√(𝟗 × 𝟖𝟏/𝟖 × 𝟏𝟐𝟓) = ³√(𝟗 × 𝟗 × 𝟗/𝟏𝟎𝟎𝟎) = 𝟗/𝟏𝟎.
EXAMPLES I. 𝑐.
If 𝑎 = 𝟖, 𝑐 = 𝟎, 𝒌 = 𝟗, 𝒙 = 𝟒, 𝒚 = 𝟏, find the value of
𝟏. √(𝟐𝑎).
𝟐. √(𝒌𝒙).
𝟑. √(𝟐𝑎𝒙).
𝟒. √(𝟐𝑎𝒌²).
𝟓. ³√(𝟑𝒌).
𝟔. ³√(𝑎𝒙³).
𝟕. ³√(𝟖𝒙³𝒚³).
𝟖. ³√(𝑐𝒚⁵).
𝟗. 𝟐𝒙√(𝟐𝑎𝒚).
𝟏𝟎. 𝟓𝒚√(𝟒𝒌𝒙).
𝟏𝟏. 𝟑𝑐√(𝒌𝒙).
𝟏𝟐. 𝟐𝒙𝒚√𝟒𝒚⁵.
𝟏𝟑. √(𝟖𝒙³/𝑎𝒌).
𝟏𝟒. √(𝟐𝟓𝑎/𝟐𝒌).
𝟏𝟓. √(𝟏𝟔𝒙/𝟒𝟗𝒚³).
𝟏𝟔. √(𝑐𝑎²/𝟏𝟔𝒌).
𝟏𝟕. ³√(𝟑𝑎/𝒌²).
𝟏𝟖. ³√(𝑎𝒙²/𝟐𝟕𝒚³).
If 𝑎 = 𝟒, 𝑏 = 𝟏, 𝑐 = 𝟐, 𝑑 = 𝟗, 𝒙 = 𝟓, 𝒚 = 𝟖, find the value of
𝟏𝟗. √𝟖𝑎𝑐.
𝟐𝟎. 𝟔√(𝟒𝑏³).
𝟐𝟏. 𝟕√(𝟓𝑑𝒙).
𝟐𝟐. √(𝑐⁵𝒚).
𝟐𝟑. √(𝟏/𝟓𝒙𝒚²).
𝟐𝟒. √(𝟏/𝟖𝑎𝑐𝑑).
𝟐𝟓. 𝟏/√(𝟗𝑎³𝑐²).
𝟐𝟔. 𝟏/√(𝟓𝑐⁴𝒙).
𝟐𝟕. ³√(𝟏/𝟒𝑐𝒙³).
𝟐𝟖. 𝟏/³√(𝟖𝑎𝑏²𝑐).
𝟐𝟗. ³√(𝟑𝒙³/𝑑²𝒚).
𝟑𝟎. ³√(𝟓𝑏⁸/𝒙⁴𝒚).
𝟑𝟏. √𝑑ᶜ.
𝟑𝟐. √𝒚ᵃ.
𝟑𝟑. √𝑏ᵈ.
𝟑𝟒. √𝑑ᵇ.
𝟏𝟓. We now proceed to find the numerical value of expressions which contain more than one term. In these each term can be dealt with singly by the rules already given, and by combining the terms the numerical value of the whole expression is explained.
𝟏𝟔. We have already, in Art. 𝟖, called attention to the importance of carefully distinguishing between coefficient and index; confusion between these is such a fruitful source of error with beginners that it may not be unnecessary once more to dwell on the distinction.
Ex. 𝟏. When 𝑐 = 𝟓, find the value of 𝑐⁴ ― 𝟒𝑐 + 𝟐𝑐³ ― 𝟑𝑐².
Here 𝑐⁴ = 𝟓⁴ = 𝟓 × 𝟓 × 𝟓 × 𝟓 = 𝟔𝟐𝟓;
𝟒𝑐 = 𝟒 × 𝟓 = 𝟐𝟎;
𝟐𝑐³ = 𝟐 × 𝑐³ = 𝟐 × 𝟓 × 𝟓 × 𝟓 = 𝟐𝟓𝟎;
𝟑𝑐² = 𝟑 × 𝑐² = 𝟑 × 𝟓 × 𝟓 = 𝟕𝟓.
Hence the value of the expression = 𝟔𝟐𝟓 — 𝟐𝟎 + 𝟐𝟓𝟎 — 𝟕𝟓 = 𝟕𝟖𝟎.
Ex. 𝟐. When 𝒑 = 𝟗, 𝒓 = 𝟔, 𝒌 = 𝟒, find the value of
𝟏/𝟑 ³√(𝒑𝒓/𝒌²) + √(𝟑𝒌 + 𝒌³ + 𝟓) ― 𝟐𝒓²/𝟗𝒌.
𝟏/𝟑 ³√(𝒑𝒓/𝒌²) + √(𝟑𝒌 + 𝒌³ + 𝟓) ― 𝟐𝒓²/𝟗𝒌 = 𝟏/𝟑 ³√(𝟓𝟒/𝟏𝟔) + √(𝟏𝟐 + 𝟔𝟒 + 𝟓) ― 𝟐×𝟑𝟔/𝟗×𝟒.
= 𝟏/𝟑 ³√(𝟐𝟕/𝟖) + √(𝟖𝟏) ― 𝟐
= 𝟏/𝟑 × 𝟑/𝟐 + 𝟗 ― 𝟐 = 𝟕 𝟏/𝟐.
𝟏𝟕. By Art. 𝟏𝟑 any term which contains a zero factor is itself zero, and may be called a zero term.
Ex. 𝟏. If 𝑎 = 𝟐, 𝑏 = 𝟎, 𝒙 = 𝟑, 𝒚 = 𝟏, find the value of
𝟒𝑎³ ― 𝑎𝑏³ + 𝟐𝒙𝒚² + 𝟑𝑎𝑏𝒙.
The expression = (𝟒 × 𝟐³) ― 𝟎 + (𝟐 × 𝟑 × 𝟏) + 𝟎.
𝟑𝟐 ― 𝟎 + 𝟔 + 𝟎 = 𝟑𝟖.
Note. The two zero terms do not affect the result.
Ex. 𝟐. Find the value of 𝟑/𝟓 𝒙² ― 𝑎²𝒚 + 𝟕𝑎𝑏𝒙 ― 𝟓/𝟐 𝒚³, when
𝑎 = 𝟓, 𝑏 = 𝟎, 𝒙 = 𝟕, 𝒚 = 𝟏.
𝟑/𝟓 𝒙² ― 𝑎²𝒚 + 𝟕𝑎𝑏𝒙 ― 𝟓/𝟐 𝒚³ =
𝟑/𝟓 (𝟕)² ― (𝟓)² × (𝟏) + 𝟕 × (𝟓) × (𝟎) × (𝟕) ― 𝟓/𝟐 × 𝟏³ =
𝟏𝟒𝟕/𝟓 ― 𝟐𝟓 + 𝟎 ― 𝟓/𝟐 = 𝟏𝟒𝟕/𝟓 ― 𝟐𝟓/𝟏 ― 𝟓/𝟐.
𝐿.𝐶.𝑀. 𝑜𝑓 𝑑𝑒𝒏𝑜𝒎𝒊𝒏𝑎𝒕𝑜𝒓𝒔 (𝟓×𝟏×𝟐) = 𝟏𝟎.
∴ 𝟏𝟒𝟕×𝟐/𝟓×𝟐 ― 𝟐𝟓×𝟏𝟎/𝟏×𝟏𝟎 ― 𝟓×𝟓/𝟐×𝟓 = 𝟐𝟗𝟒/𝟏𝟎 ― 𝟐𝟓𝟎/𝟏𝟎 ― 𝟐𝟓/𝟏𝟎 = 𝟐𝟗𝟒―𝟐𝟓𝟎―𝟐𝟓/𝟏𝟎 = 𝟏𝟗/𝟏𝟎 ∨ 𝟏 𝟗/𝟏𝟎.
Note. The zero term does not affect the result.
𝟏𝟖. In working examples the student should pay attention to the following hints:
𝟏. Too much importance cannot be attached to neatness of style and arrangement. The beginner should remember that neatness is in itself conducive to accuracy.
𝟐. The sign = should never be used except to connect quantities which are equal. Beginners should be particularly careful not to employ the sign of equality in any vague and inexact sense.
𝟑. Unless the expressions are very short the signs of equality in the steps of the work should be placed one under the other.
𝟒. It should be clearly brought out how each step follows from the one before it ; for this purpose it mil sometimes be advisable to add short verbal explanations; the importance of this will be seen later.
EXAMPLES I. 𝑑.
If 𝑎 = 𝟐, 𝑏 = 𝟑, 𝑐 = 𝟏, 𝑑 = 𝟎, find the numerical value of
𝟏. 𝟔𝑎 + 𝟓𝑏 ― 𝟖𝑐 + 𝟗𝑑.
𝟐. 𝟑𝑎 ― 𝟒𝑏 + 𝟔𝑐 + 𝟓𝑑.
𝟑. 𝟔𝑎𝑏 ― 𝟑𝑐𝑑 + 𝟐𝑑𝑎 ― 𝟓𝑐𝑏 𝟐𝑑𝑏.
𝟒. 𝑎𝑏𝑐 + 𝑏𝑐𝑑 + 𝑐𝑑𝑎 + 𝑑𝑎𝑏.
𝟓. 𝟑𝑎𝑏𝑐 ― 𝟐𝑏𝑐𝑑 + 𝟐𝑐𝑑𝑎 ― 𝟒𝑑𝑎𝑏.
𝟔. 𝟐𝑏𝑐 + 𝟑𝑐𝑑 ― 𝟒𝑑𝑎 + 𝟓𝑎𝑏.
𝟕. 𝟑𝑏𝑐𝑑 + 𝟓𝑐𝑑𝑎 ― 𝟕𝑑𝑎𝑏 + 𝑎𝑏𝑐.
𝟖. 𝑎² + 𝑏² + 𝑐² + 𝑑².
𝟗. 𝟐𝑎² + 𝟑𝑏³ ― 𝟒𝑐⁴.
𝟏𝟎. 𝑎⁴ + 𝑏⁴ ― 𝑐⁴.
If 𝑎 = 𝟏, 𝑏 = 𝟐, 𝑐 = 𝟑, 𝑑 = 𝟎, find the numerical value of
𝟏𝟏. 𝑎³ + 𝑏³ + 𝑐³ + 𝑑³.
𝟏𝟐. 𝟏/𝟐 𝑏𝑐³ ― 𝑎³ ― 𝑏³ ― 𝟑/𝟒 𝑎𝑏³𝑐.
𝟏𝟑. 𝟑𝑎𝑏𝑐 ― 𝑏²𝑐 ― 𝟔𝑎³.
𝟏𝟒. 𝟐𝑎² + 𝟐𝑏² + 𝟐𝑐² + 𝟐𝑑² ― 𝟐𝑏𝑐 ― 𝟐𝑐𝑑 ― 𝟐𝑑𝑎 ― 𝟐𝑎𝑏.
𝟏𝟓. 𝑎² + 𝟐𝑏² + 𝟐𝑐² + 𝑑² + 𝟐𝑎𝑏 + 𝟐𝑏𝑐 + 𝟐/𝟕 𝑐𝑑.
𝟏𝟔. 𝟐𝑐² + 𝟐𝑎² + 𝟐𝑏² ― 𝟒𝑐𝑏 + 𝟔𝑎𝑏𝑐𝑑.
𝟏𝟕. 𝟏𝟑𝑎² + 𝟏𝟏/𝟗 𝑐⁴ + 𝟐𝟎𝑎𝑏 ― 𝟏𝟔𝑎𝑐 ― 𝟏𝟔𝑏𝑐.
𝟏𝟖. 𝟔𝑎𝑏 ― 𝟒/𝟑 𝑎𝑐² ― 𝟐𝑎 + 𝟏/𝟖 𝑏⁴ ― 𝟑𝑑 + 𝟒/𝟗 𝑐³.
𝟏𝟗. 𝟏𝟐𝟓𝑏⁴𝑐 ― 𝟗𝑑⁵ + 𝟑𝑎𝑏𝑐²𝑑.
If 𝑎 = 𝟖, 𝑏 = 𝟐, 𝑐 = 𝟏, 𝒙 = 𝟗, 𝒚 = 𝟒, find the value of
𝟐𝟎. 𝟓/𝟑 𝑎 ― 𝟏/𝟗 𝑏³ + 𝟕/𝟖 𝒚².
𝟐𝟏. 𝟓/𝟐𝟕 𝑎𝒙 ― 𝟑𝟐/𝒚² ― 𝟔𝑎/𝑐𝒙𝒚.
𝟐𝟐. 𝟑𝑎²𝑏/𝑐𝒙𝒚² ― 𝟓𝒚/𝑎.
𝟐𝟑. ³√(𝟔𝑐𝒚⁴/𝒙²) + 𝟐√(𝟑𝑎³/𝟒𝑏³).
𝟐𝟒. ³√(𝑏𝒙𝒚) ― 𝟏/𝟖 𝑏² + 𝟖𝒙²/𝑏²𝒚.
𝟐𝟓. 𝟑/𝟒 𝑎𝑐 ― √(𝑏²/𝟗𝒚) ― ³√(𝑏𝒚/𝒙²).
𝟐𝟔. 𝟓𝑏²𝒚³/𝟏𝟐𝑎²𝒙 ― ³√(𝑎𝒙⁴/𝑏²𝒚²) + √(𝑎𝑏³/𝟑𝒙).
By Sohail Tahir
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