to verify the algebraic identity: (a+b)³=a³+b³+3a²b+3ab²
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Answer:
(a+b)³ = a³ + b³ + 3a²b + 3ab²
Step-by-step explanation:
(a+b)³ = (a+b)(a+b)(a+b)
(a+b)³ = (a² + ab + ab + b²)(a+b)
(a+b)³ = (a² + 2ab + b²)(a+b)
(a+b)³ = (a³ + a²b + 2a²b + 2ab² + ab² + b³)
(a+b)³ = a³ + b³ + 3a²b + 3ab²
Answered by
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Answer:
OBJECTIVE
To verify the algebraic identity (a+b)³ = a³+b³+ 3a²b + 3ab².
Materials Required
Acrylic sheets
Adhesive/Adhesive tape
Scissors
Geometry Box
Cutter
Prerequisite Knowledge
Concept of cuboid and its volume.
Concept of cube and its volume.
Theory
Cuboid A cuboid is a solid bounded by six rectangular plane surfaces, e.g. Match box, brick, box, etc., are cuboid, (see Fig. 7.1)
NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² 1
Properties of cuboid are
In a cuboid, there are 6 faces, 12 edges and 8 corners (four at bottom and four at top face) which are called vertices.
Opposite faces of a cuboid are equal and parallel.
The line segment joining the opposite vertices of cuboid is called the diagonal of a cuboid.
There are four diagonals in a cuboid which are equal in length.
Volume of cuboid = lbh
where, l = length, b = breadth and h = height
Cube A cuboid whose length, breadth and height are same, is called a cube, (see Fig. 7.2)
NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² 2
Properties of cube are
In a cube, there are 6 faces, 12 edges and 8 corners (four at bottom and four at top face) which are called vertices.
All the six faces of a cube are congruent square faces.
Each edge of a cube have same length.
Volume of cube = a³
where, a is side of cube.
Procedure
Cut six squares of equal side a units from acrylic sheet. Paste all of them to form a cube by using adhesive tape/adhesive, (see Fig. 7.3)
NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² 3
Cut six squares of equal side b units (b < a) from acrylic sheet. Paste all of them to form a cube by using adhesive tape/adhesive, (see Fig. 7.4)
NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² 4
Also, cut 12 rectangles of length b units and breadth a units and 6 squares of side a units. Paste all of them to form a cuboid, (see Fig. 7.5)
NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² 5
Cut 12 rectangles of length a units and breadth b units and 6 squares of side b units. Paste all of them to form a cuboid, (see Fig. 7.6)
NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² 6
Arrange the cubes obtained in Fig 7.3 and Fig 7.4 and the cuboids obtained in Fig 7.5 and Fig 7.6 as shown in Fig 7.7
NCERT Class 9 Maths Lab Manual - Verify the Algebraic Identity (a+b)³ = a³+b³+ 3a²b + 3ab² 7
Demonstration
For Fig. 7.3, volume of cube of side a units = a³
For Fig. 7.4, volume of cube of side b units = b³
For Fig. 7.5, volume of a cuboid of dimensions a x a x b units = a²b
So, volume of all three such cuboids = a²b + a²b + a²b = 3a²b
For Fig. 7.6, volume of a cuboid of dimensions a x b x b units = ab²
So, volume of all three such cuboids = ab² + ab² + ab² = 3ab²
In Fig. 7.7, we have obtained the cube of side (a + b) units.
So, volume of cube = (a + b)³
As, volume of cube of Fig. 7.7 = (Volume of cube of Fig. 7.3) + (Volume of cube of Fig. 7.4) + (Volume of three cuboids of Fig. 7.5) + (Volume of three cuboids of Fig. 7.6)
=> (a+b)³ = a³+b³+ 3a²b + 3ab²
Here, volume is in cubic units.
Observation
On actual measurement, we get
a =…….. , b = …….. ,
So, a³ =…….. , b³ = …….. ,
a2b = …….. , 3a²b = …….. ,
ab² = …….. , 3ab² = …….. ,
(a + b)³ = …….. ,
Hence, (a+b)³ = a³+b³+ 3a²b + 3ab²
Result
The algebraic identity (a+b)³ = a³+b³+ 3a²b + 3ab² has been verified.
Application
The identity is useful for
calculating the cube of a number which can be expressed as the sum of two convenient numbers.
simplification and factorization of algebraic expressions
Step-by-step explanation:
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