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Simplification Of Expressions (multiple Operations And Multiple Variables)

Simplification of Expressions involving Addition and Subtraction

  • Simplification means making things easier.
  • We can simplify the expressions having various operations.
  • Here, we will learn to simplify the expressions having two operations i.e., addition and subtraction.

For example: 

Simplify: \(8x+16y-2z+3x-7y+5z\)

First, arrange the like terms together.

\(8x+3x+16y-7y-2z+5z\)

Now, take the common variables out.

\(=x(8+3)+y(16-7)+z(-2+5)\)

\(=x(11)+y(9)+z(3)\)

\(=11x+9y+3z\)

Illustration Questions

Simplify the expression: \(10x+7y-3a+2a+8x-3y+10\)

A \(x+y-a\)

B \(4x+18y-a\)

C \(18x+4y-a+10\)

D \(x+y+2a\)

×

Given expression: 

\(10x+7y-3a+2a+8x-3y+10\)

First, arrange the like terms together and take the common variables out.

Thus, \(10x+7y-3a+2a+8x-3y+10\)

\(= 10x+8x+7y-3y-3a+2a+10\)

\(=x(10+8)+y (7-3) + a (-3+2)+10\)

\(=x(18) + y (4)+a(-1)+10\)

\(=18x+4y-a+10\)

Hence, option (C) is correct.

Simplify the expression: \(10x+7y-3a+2a+8x-3y+10\)

A

\(x+y-a\)

.

B

\(4x+18y-a\)

C

\(18x+4y-a+10\)

D

\(x+y+2a\)

Option C is Correct

Simplification of Expressions involving Addition and Multiplication

  • Here, we will learn to simplify the expressions involving addition and multiplication.

For example: 

Simplify: \(2×a×b×c\;+\;5×a×b×c\;+\;7×a×a×a\)

We will take the common variables out.

\(2×a×b×c\;+\;5×a×b×c\;+\;7×a×a×a\)

\(= 2abc\;+\; 5abc \; +\; 7a^3\)

\(=abc (2+5)+7a^3\)

\(=abc (7)+7a^3\)

\(=7abc+7a^3\)

Illustration Questions

Simplify the expression: \(3×e×e×f\;+\;5×f×f×e\; +\; 8×e×f×e\;+\; 5×e×f×f\)

A \(11e^2f+10f^2e\)

B \(10e^2f+11f^2e\)

C \(7e^2+5f^2\)

D \(2e^2f\)

×

Given expression:

\(3×e×e×f\;+\;5×f×f×e\; +\; 8×e×f×e\;+\; 5×e×f×f\)

First, arrange the like terms together. Next, count the number of times each variable is multiplied by itself and write it as the power of that variable and then perform the operation.

Thus, \(3×e×e×f\;+\;5×f×f×e\; +\; 8×e×f×e\;+\; 5×e×f×f\)

\(=3×e×e×f+8×e×e×f+5×f×f×e+5×e×f×f\)

\(=3e^2f+8e^2f+5f^2e+5f^2e\)

\(=e^2f(3+8)+f^2e(5+5)\)

\(=e^2f(11)+f^2e(10)\)

\(=11e^2f+10f^2e\)

Hence, option (A) is correct.

Simplify the expression: \(3×e×e×f\;+\;5×f×f×e\; +\; 8×e×f×e\;+\; 5×e×f×f\)

A

\(11e^2f+10f^2e\)

.

B

\(10e^2f+11f^2e\)

C

\(7e^2+5f^2\)

D

\(2e^2f\)

Option A is Correct

Simplification of Expressions involving Subtraction and Multiplication

  • Here, we will learn to simplify the expressions involving subtraction and multiplication.

For example: 

Simplify: \(4×a×b×c-8×a×b×c-3×a×a×b\)

We will take the common variables out from the expression.

\(4×a×b×c-8×a×b×c-3×a×a×b\)

\(=4abc-8abc-3a^2b\)

\(=abc(4-8)-3a^2b\)

\(=abc(-4)-3a^2b\)

\(=-4abc-3a^2b\)

Illustration Questions

Simplify the expression: \(-[7(a-2b)-4b]\)

A \(-18a-7b\)

B \(7b+18a\)

C \(18b+7a\)

D \(-7a+18b\)

×

Given expression:

\(-[7(a-2b)-4b]\)

Here, first, we will simplify the inner parenthesis and then the outer parenthesis. Later we will take the common variables out.

Thus, \(-[7(a-2b)-4b]\)

\(=-[7a-14b-4b]\)

\(=-[7a-b(14+4)]\)

\(=-[7a-b(18)]\)

\(=-[7a-18b]\)

\(=-7a+18b\)

Hence, option (D) is correct.

Simplify the expression: \(-[7(a-2b)-4b]\)

A

\(-18a-7b\)

.

B

\(7b+18a\)

C

\(18b+7a\)

D

\(-7a+18b\)

Option D is Correct

Simplification of Expressions involving Addition and Division

  • Here, we will learn to simplify the expressions involving addition and division.

For example: 

Simplify: \(\dfrac{6p^3q^2+10p^2q}{4q}\)

First, we take the commons out and then divide.

\(\dfrac{6p^3q^2+10p^2q}{4q}\)

\(=\dfrac{ \not{2}p^2\not{q}(3pq+5)}{ \not{4}^2 \not{q}}\)

\(=\dfrac{p^2(3pq+5)}{2}\)

\(=\dfrac{3p^3q+5p^2}{2}\)

Illustration Questions

Simplify the expression: \(\dfrac{7ab^2c+8a^2bc}{abc}\)

A \(7ab\)

B \(7b+8a\)

C \(7b-8a\)

D \(7ac\)

×

Given expression:

\(\dfrac{7ab^2c+8a^2bc}{abc}\)

First, we take the common variables out and then divide.

Thus, \(\dfrac{7ab^2c+8a^2bc}{abc}\)

\(=\dfrac{ abc(7b+8a)}{ abc}\;= 7b+8a\)

Hence, option (B) is correct. 

Simplify the expression: \(\dfrac{7ab^2c+8a^2bc}{abc}\)

A

\(7ab\)

.

B

\(7b+8a\)

C

\(7b-8a\)

D

\(7ac\)

Option B is Correct

Simplification of Expressions involving Subtraction and Division

  • Here, we will learn to simplify the expressions involving subtraction and division.

For example: 

Simplify: \(\dfrac{3ab^2c-6b^2c^2-3ab^2c^2}{3b^2c}\)

First, we take the commons out and then divide. 

\(\dfrac{3ab^2c-6b^2c^2-3ab^2c^2}{3b^2c}\)

\(=\dfrac{\not 3 \not b^2 \not c(a-2c-ac)}{\not 3 \not b^2 \not c}\)

\(=a-2c-ac\)

\(=a-c(2+a)\)

Illustration Questions

Simplify the expression: \(\dfrac{abc-2a^2b^2c}{2a^2b^2c-abc}\)

A \(2a^2b^2c\)

B \(abc\)

C \(1\)

D \(-1\)

×

Given expression:

\(\dfrac{abc-2a^2b^2c}{2a^2b^2c-abc}\)

First, we take the common variables out and then divide. 

Thus, \(\dfrac{abc-2a^2b^2c}{2a^2b^2c-abc}\)

\(=\dfrac{\not a \not b \not c(1-2ab)}{\not a \not b \not c(2ab-1)}\)

\(=\dfrac{-(2ab-1)}{(2ab-1)}=-1\)

Hence, option (D) is correct.

Simplify the expression: \(\dfrac{abc-2a^2b^2c}{2a^2b^2c-abc}\)

A

\(2a^2b^2c\)

.

B

\(abc\)

C

\(1\)

D

\(-1\)

Option D is Correct

Simplification of Expressions involving Multiplication and Division

  • Here, we will learn to simplify the expressions involving multiplication and division.

For example: \(\dfrac{6×a×a×a×b×b×c}{3×a×b×c}\)

We will cancel out the same variables appearing in both numerator and denominator.

\(\dfrac{ \not{6}^2× \not{a}×a×a×b× \not{b}× \not{c}}{ \not{3}× \not{a}× \not{b}× \not{c}}\)

\(= 2×a×a×b\)

\(=2a^2b\)

Illustration Questions

Simplify the expression: \(\dfrac{16×p×q×r}{8×p×p×r×q×q×q}\)

A \(\dfrac{2}{pq^2}\)

B \(2pq^2\)

C \(pq^2\)

D \(2\)

×

Given expression:

\(\dfrac{16×p×q×r}{8×p×p×r×q×q×q}\)

We will cancel out the same variables appearing in both numerator and denominator.

Thus, \(\dfrac{16× \not{p}× \not{q}× \not{r}}{8×p× \not{p}× \not{r}× \not{q}×q×q}\)

\(=\dfrac{2}{p×q×q}\)

\(=\dfrac{2}{pq^2}\)

Hence, option (A) is correct.

Simplify the expression: \(\dfrac{16×p×q×r}{8×p×p×r×q×q×q}\)

A

\(\dfrac{2}{pq^2}\)

.

B

\(2pq^2\)

C

\(pq^2\)

D

\(2\)

Option A is Correct

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