Factorise [tex]$125 a^3-64 b^3$[/tex].
Answer (2)
Recognize the expression as a difference of cubes: ( 5 a ) 3 − ( 4 b ) 3 .
Apply the difference of cubes factorization formula: A 3 − B 3 = ( A − B ) ( A 2 + A B + B 2 ) , where A = 5 a and B = 4 b .
Substitute A = 5 a and B = 4 b into the formula and simplify.
The factorised form is ( 5 a − 4 b ) ( 25 a 2 + 20 ab + 16 b 2 ) .
Explanation
Recognizing the Difference of Cubes We are asked to factorise the expression 125 a 3 − 64 b 3 . This expression is a difference of cubes. We can rewrite the expression as ( 5 a ) 3 − ( 4 b ) 3 .
Stating the Formula The difference of cubes factorization formula is given by A 3 − B 3 = ( A − B ) ( A 2 + A B + B 2 ) . In our case, A = 5 a and B = 4 b .
Applying the Formula and Simplifying Substituting A = 5 a and B = 4 b into the formula, we get: ( 5 a ) 3 − ( 4 b ) 3 = ( 5 a − 4 b ) (( 5 a ) 2 + ( 5 a ) ( 4 b ) + ( 4 b ) 2 ) Simplifying the terms inside the second parenthesis: ( 5 a ) 2 = 25 a 2 ( 5 a ) ( 4 b ) = 20 ab ( 4 b ) 2 = 16 b 2 So, the expression becomes: ( 5 a − 4 b ) ( 25 a 2 + 20 ab + 16 b 2 )
Final Factorised Form Therefore, the factorised form of 125 a 3 − 64 b 3 is ( 5 a − 4 b ) ( 25 a 2 + 20 ab + 16 b 2 ) .
Examples
The difference of cubes factorization is useful in simplifying algebraic expressions and solving equations. For example, consider designing a storage container in the shape of a cube. If you want to increase the volume of the container from 64 cubic units to 125 cubic units, you can use the difference of cubes to determine the change in the side length required. This type of problem arises in engineering and design when dealing with volumes and dimensions.
The expression 125 a 3 − 64 b 3 can be factored as ( 5 a − 4 b ) ( 25 a 2 + 20 ab + 16 b 2 ) using the difference of cubes formula. By substituting A = 5 a and B = 4 b , we simplify the expression. The final factorized form is therefore ( 5 a − 4 b ) ( 25 a 2 + 20 ab + 16 b 2 ) .
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