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Simplification Of Expressions (multiple Operations Single Variable)

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: \(5a+6a-7a+8a\)

Here, we know that \(5a,\,6a,\,-7a\) and \(8a\) are all like terms.

We can simplify the expression by taking the common variable out.

Thus, \(5a+6a-7a+8a\)

\(= a(5+6-7+8)\)

\( = a (19-7)\)

\( = a (12)\)

\( = 12a\)

Illustration Questions

Simplify the expression: \(7z+2z-3z-2z\)

A \(2z\)

B \(7z\)

C \(4z\)

D \(3z\)

×

Given expression:

\(7z + 2z-3z-2z\)

We can simplify the expression by taking the common variable out.

Thus, \(7z+2z-3z-2z\)

\( =z(7+2-3-2)\)

\(= z (4)\)

\( = 4z\)

Hence, option (C) is correct.

Simplify the expression: \(7z+2z-3z-2z\)

A

\(2z\)

.

B

\(7z\)

C

\(4z\)

D

\(3z\)

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: \(5× a×a+6×5×a\)

Here, in both the terms, \(5\) and \(a\) are common.

Thus, we can simplify the expression by taking the commons out i.e., \(5\) and \(a\).

\(5×a×a + 6× 5 ×a\)

\( = 5a (a+6)\)

Illustration Questions

Simplify the expression: \(2 × b×b+3×b×b×b\)

A \(4b(2+3b)\)

B \(3b(b+2b)\)

C \(b(2+3b)\)

D \(b^2 (2+3b)\)

×

Given expression:

\(2× b×b + 3 × b× b×b\)

Here, in both the terms, two b's are common.

Thus, we can simplify the expression by taking the commons out i.e., \(b×b\).

Thus, \(2×b×b + 3× b×b×b\)

\( = b×b (2+3×b)\)

\( = b^2 (2+3b)\)

Hence, option (D) is correct.

Simplify the expression: \(2 × b×b+3×b×b×b\)

A

\(4b(2+3b)\)

.

B

\(3b(b+2b)\)

C

\(b(2+3b)\)

D

\(b^2 (2+3b)\)

Option D is Correct

Simplification of Expressions involving Subtraction and Multiplication

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

For example: 

Simplify: \(6×b×b×b - 6×b×b\)

Here, \(6\) and two b's are common in both the terms.

Thus, we can simplify the expression by taking the commons out i.e., \(6\) and two b's.

\(6×b×b×b-6×b×b\)

\( = 6×b×b(b-1)\)

\( = 6b^2(b-1)\)

Illustration Questions

Simplify the expression: \(3×c×c-6×c\)

A \(3c(c-2)\)

B \(c-2\)

C \(3c\)

D \(6c\)

×

Given expression:

\(3×c×c - 6×c\)

Here, \(3\) and \(c\) are common in both the terms.

Thus, we can simplify the expression by taking the commons out i.e., \(3\) and \(c\).

Thus, \(3×c×c-6×c\)

\(= 3×c(c-2)\)

\(= 3c (c-2)\)

Hence, option (A) is correct.

Simplify the expression: \(3×c×c-6×c\)

A

\(3c(c-2)\)

.

B

\(c-2\)

C

\(3c\)

D

\(6c\)

Option A is Correct

Simplification of Expressions involving Addition and Division

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

For example:

Simplify: \(\dfrac{6× d×d}{3d} + \dfrac{10×d×d×d}{5d×d}\)

In expressions involving division, similar variables in the numerator and the denominator get canceled by each other i.e.,

\(\dfrac{ \not{6}^2× \not{d}×d}{ \not{3}× \not{d}} + \dfrac{\not10^2×d× \not{d}× \not{d}}{\not5× \not{d}× \not{d}}\)

\( = 2d+2d\)

\( = 2d(1+1)\)  [Taking the commons out i.e, 2d]

\(= 2d(2)\)

\(= 4d\)

Illustration Questions

Simplify the expression: \(\dfrac{2×e×e}{2×e} + \dfrac{6×e×e×e}{3×e}\)

A \(2e\)

B \(e (1+2e)\)

C \(e\)

D \(1+2e\)

×

Given expression:

\(\dfrac{2×e×e}{2×e} + \dfrac{6×e×e×e}{3×e}\)

In expressions involving division, similar variables in the numerator and the denominator get canceled by each other i.e.,

\(\dfrac{ \not{2}× \not{e}×e}{ \not{2}× \not{e}} + \dfrac{ \not{6}^2× \not{e}×e×e}{ \not{3}× \not{e}}\)

\( =e + 2e×e\)

\( = e (1+2e)\)   [\(e\) is common in both the terms]

Hence, option (B) is correct.

Simplify the expression: \(\dfrac{2×e×e}{2×e} + \dfrac{6×e×e×e}{3×e}\)

A

\(2e\)

.

B

\(e (1+2e)\)

C

\(e\)

D

\(1+2e\)

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: \(\dfrac{4×e×e}{2×e} - \dfrac{9×e×e×e}{3×e×e}\)

In expressions involving division, similar variables in the numerator and the denominator get canceled by each other i.e.,

\(\dfrac{ \not{4}^2 × \not{e}×e}{ \not{2}× \not{e}} - \dfrac{ \not{9}^3× \not{e}× \not{e}×e}{ \not{3}× \not{e}× \not{e}}\)

\( = 2e-3e\)

\(=e (2-3)\)  [Taking \(e\) as common]

\(=e (-1)\)

\(= -e\)

Illustration Questions

Simplify the expression: \(\dfrac{12×z×z×z}{3×z} - \dfrac{6×z×z}{z}\)

A \(z-3\)

B \(2z-3\)

C \(2z\)

D \(2z(2z-3)\)

×

Given expression:

\(\dfrac{12×z×z×z}{3×z} - \dfrac{6×z×z}{z}\)

In expressions involving division, similar variables in the numerator and the denominator get canceled by each other i.e.,

\(\dfrac{\not12^4× \not{z}×z×z}{\not3× \not{z}} - \dfrac{6× \not{z}×z}{ \not{z}}\)

\(= 4 × z×z- 6×z\)

\( = 2×z (2×z-3)\)  [Taking \(2z\) as common]

\( = 2z (2z-3)\)

 

Hence, option (D) is correct.

Simplify the expression: \(\dfrac{12×z×z×z}{3×z} - \dfrac{6×z×z}{z}\)

A

\(z-3\)

.

B

\(2z-3\)

C

\(2z\)

D

\(2z(2z-3)\)

Option D is Correct

Multiplication and Division

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

For example: 

Simplify: \(\dfrac{3×z×5×z×z×z}{z×3×5×z}\)

In expressions involving division, similar variables in the numerator and the denominator get canceled by each other i.e.,

\(\dfrac{\not {3}×\not {z}×\not {5}×\not {z}×z×z}{\not {z}×\not {3}×\not {5}×\not {z}}\)

\(=z×z\)

\(=z^2\)

Illustration Questions

Simplify the expression: \(\dfrac{10×a×a×a×5}{a×5×a}\)

A \(2a\)

B \(5a\)

C \(10a\)

D \(12a\)

×

Given expression:

\(\dfrac{10×a×a×a×5}{a×5×a}\)

In expressions involving division, similar variables in the numerator and the denominator get canceled by each other i.e.,

\(\dfrac{10×a×\not {a}×\not {a}×\not {5}}{\not {a}×\not {5}×\not {a}}\)

\( = 10a\)

 

Hence, option (C) is correct.

Simplify the expression: \(\dfrac{10×a×a×a×5}{a×5×a}\)

A

\(2a\)

.

B

\(5a\)

C

\(10a\)

D

\(12a\)

Option C is Correct

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