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Simplify (2/(3k)) + (k/(k + 1)) divided by (k/(k + 1)) – (3/k)

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  • Last Updated : 21 Nov, 2021

Algebra is the branch of mathematics in which we study to find the value of unknowns. It consists of numerals, variables, and fundamental arithmetic operators. The terms which have constant value are known as the numerals and it is represented by numbers, the terms that do not have constant value are known as the variable and is represented by letters or symbols. Algebra is basically used to formalize the different formulas in mathematics with the help of variables or symbols.

Algebraic Expression

An algebraic expression is the systematic representation of numerals, variables with operators. It is basically a representation of mathematical statements into a mathematical expression. 

For example, ‘Three times a number is subtracted from 21′ can be written as ’21 – 3x’. Here we don’t know the number so we represent it by x. The negative sign separates the expression into two terms. So on the basis of the number of terms the expression can be classified into the following types.

  • Monomial: If the number of terms in an expression is one then it is known as a monomial. For example, 9t, 6y, etc
  • Binomial: If the number of terms in an expression is two then it is known as binomial expression. Example: 8x-9y, 8t-6u, etc.
  • Trinomial: If the number of terms in an expression is three then it is known as trinomial. Example: 8a+3b+5c, 8e-6g-6s, etc.
  • Polynomial: If the number of terms in an expression is one or more than one then it is known as a polynomial.

Like terms and unlike terms

If the variable terms of an expression are the same then it is known as the like terms of an algebraic expression and if variable terms are not the same then it is known as unlike terms.

For example: 9x² – 6y +3x -5x² +4y – 9

In the above expression, 9x² and 5x² have the same variable, 6y and 4y have the same variable. So these terms are like terms.

Simplify (2/(3k)) + (k/(k+ 1)) divided by (k/(k + 1)) – (3/k)

Solution:

Step to solve the problem:

Step 1: Write the given mathematical statement in expression with the help of numerals, variables, and operators accordingly.

= {(2/(3k) + (k/(k+1))} ÷ {(k/(k + 1)) – (3/k)}

Step 2: Simplify the bracket part first. If the denominator of the fraction is not the same then do the cross multiplication and simply.

= [{(2×(k+1) + (k×3k)}/(3k)×(k+1)] ÷ [{(k×k) – 3×(k+1)}/k×(k+1)]

= {(2k+2+3k²)/3k×(k+1)} ÷ {(k²-3k-3)/k×(k+1)}

Step 3: If a fraction is divided by another fraction then it can be written as:

(a/b)÷(c/d)=(a×d)/(b×c)

= {(2k+2+3k²)×(k×(k+1))/3k×(k+1)×((k²-3k-3)}

Step 4: Cancel to the common factor of the numerator and denominator.

= (3k²+2k+2)/(k²-3k-3)

This expression can not further simplified. So the simplification of  (2/(3k)) + (k/(k+ 1)) divided by (k/(k + 1)) – (3/k) is (3k²+2k+2)/(k²-3k-3).

Similar Questions

Question 1: Simplify (2/(5x)) + (x/(x+ 3)) divided by (x/(x + 3)) – (6/x).

Solution:

Write the given mathematical statement in expression with the help of numerals, variables, and operators accordingly.

= {(2/(5x)) + (x/(x+ 3))} ÷ {(x/(x + 3)) – (6/x)}

Simplify the bracket part first. If the denominator of the fraction is not the same then do the cross multiplication and simply.

= {(2×(x+3)+5x×x)/(5x×(x+3)} ÷ {(x×x -6×(x+3))/(x×(x+3))}

= {(2x+6+5x²)/(5x×(x+3))} ÷ {(x²-6x-18)/(x×(x+3))}

If a fraction is divided by another fraction then it can be written as: (a/b)÷(c/d)=(a×d)/(b×c)

= {(2x+6+5x²)×(x×(x+3))}/{(5x×(x+3)×(x²-6x-18)}

Cancel to the common factor of the numerator and denominator.

= (5x²+2x+6)/(5×(x²-6x-18))

= (5x²+2x+6)/(5x²-30x-90)

So the simplification of (2/(5x)) + (x/(x+ 3)) divided by (x/(x + 3)) – (6/x) is (5x²+2x+6)/(5x²-30x-90).

Question 2: Simplify 3a/(a+2)-1 divided by (a×(a+6))/(a+2)+2.

Solution:

Write the given mathematical statement in expression with the help of numerals, variables, and operators accordingly.

= {3a/(a+2)-1} ÷ {(a×(a+6))/(a+2)+2}

Simplify the bracket part first. If the denominator of the fraction is not the same then do the cross multiplication and simply.

= {(3a-a-2)/(a+2)} ÷ {(a²+6a+2a+4)/(a+2)

= {(2a-2)/(a+2)} ÷ {(a²+8a+4)/(a+2)}

= {2(a-1)/(a+2)} ÷ {(a²+8a+4)/(a+2)}

If a fraction is divided by another fraction then it can be written as: (a/b)÷(c/d) = (a×d)/(b×c)

= {2×(a-1)×(a+2)}/{(a+2)×(a²+8a+4)}

Cancel to the common factor of the numerator and denominator.

= 2(a-1)/(a²+8a+4)

So the simplification of 3a/(a+2)-1 divided by (a×(a+6))/(a+2)+2 is 2(a-1)/(a²+8a+4)

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