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Generally, Algebraic expressions are the symbol or a combination of symbols used in algebra containing one or more numbers, variables, and arithmetic operations. 2xyz, 2x +yz, 2x + y, etc. are the examples of Algebraic Expressions.
Evaluation of Algebraic Expressions
When we replace the variable of a term or expression with numbers, the value of the terms or expression is obtained. It is called the evaluation of a term or expression. For examples,
If x = 3, y = 4 and z = 2, then
2xyz = 2 × 3 × 4 × 2 = 48
Law of Indices
While performing the operations of multiplication and division of algebraic expression it needs to work out indices of the same bases under the certain rules. These are called the law of indices.
Addition and Subtraction of algebraic expression
To add and subtract the algebraic expression, there should be the like and unlike terms. Unlike terms are those which do not have the same base and like terms are those which have the same base. While adding and subtracting algebraic term, we should add or subtract the coefficients of
like terms. For example,
Multiplication of algebraic expressions
When the coefficients of the terms of the terms are multiplied and the power of the same bases are added then it is called Multiplication of algebraic expressions. For example,
Multiplication of polynomials by monomials
In this case, in each term of polynomials is separately multiplied by the monomial. For example
Multiply, (b+c) by x.
Here, x× (b+c) = bx +cx
Multiplication of polynomials
In this case, each term of polynomials is separately multiplied by each term of another polynomial. Then, the product is simplified. For examples,
(a+b) by (x+y)
Here, (x+y)(a+b)
or, x(a+b) = y(a+b)
or, ax+bx+ay+by
Some special products formulae
Division of algebraic expressions
While dividing a monomial by another monomial, divide the coefficient of dividend by the coefficient of of divisor. Then substract the power of the base of divisor from the power of the same base of dividend. For examples,
18x^{4}y^{3} by 6x^{2}y^{2}
or, 18x^{4}y^{3}÷ 6x^{2}y^{2} = \(\frac{18x^4y^3}{6x^2y^2}\)
or, 3x^{4-2}y^{3-2} = 3x^{2}y
Division of polynomials by monomials
In this case each term of a polynomial is separately dividend by the monomial. For example,
(12x^{4} - 15x^{3})÷ 3x^{2}
= \(\frac{24x^4}{3x^2}\) - \(\frac{15x^3}{3x^2}\)
= 4x^{4-2} - 5x^{3-2}
= 4x^{2} - 5x
Division of polynomials by polynomials
In this case, at first we should arrange the terms of divisor and dividend in descending or ascending order of power of common bases. Then we should atart the division dividing the term of dividend with the highest power.
Divisor | Dividend |
2 | 1296 |
2 | 648 |
2 | 324 |
2 | 162 |
3 | 81 |
3 | 27 |
3 | 9 |
3 |
\(\therefore\) 1296 = 2×2×2×2×3×3×3×3
= 2^{4}×3^{4}
=(2×3)^{4}
= 6^{4 } ans.
Solution:
(\(\frac{16}{81}\))^{\(\frac{1}{4}\)}
= (\(\frac{2^4}{3^4}\))^{\(\frac{1}{4}\)}
= (\(\frac{2}{3}\))^{4×\(\frac{1}{4}\)}
= (\(\frac{2}{3}\))^{1}
= \(\frac{2}{3}\)
Solution:
= (4x - 7) (2x^{2} + 3x - 5)
= 4x(2x^{2} + 3x - 5) - 7 (2x^{2} + 3x - 5)
= 8x^{3} + 12x^{2} - 20x - 14x^{2} - 21x + 35
= 8x^{3} - 2x^{2} - 41x + 35
Solution:
Here, (a+b) = 2
\(\therefore\) (a+b)^{3} = 2^{3}
or, a^{3} + 3a^{2}b + 3ab^{2} + b^{3} = 8
or, a^{3} + b^{3} + 3ab^{2} = 8
or, a^{3} + b^{3} + 6ab = 8
So, the required value of a^{3} + b^{3} + 6ab is 8.
If x = 5, y = 4 and z = 2, then evaluate the xy(-z)^{3}.
If x = 2 and y = 3 what will be the value of y^{x}.
What should be added to 7x to get 12 x?
What should be subtracted from 5pq to get 10pq?
What will be the final products of (a - b)^{3}.
If (a + b) = 2, what will be the valuee of a^{3} + b^{3} + 6ab
What will be the quotients of 6x^{2} ÷ 3x.
If p -(frac{1}{p}) = 8 find the value of p^{2} + (frac{1}{p^2}).
Find the squares of the (y - 3).
Find the area of a square when l = 4.6 cm.
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