Factorise: [tex]$16 x^{23}-64 x^{50}$[/tex]
Answer (2)
Find the greatest common factor (GCF) of the coefficients: GCF(16, 64) = 16.
Find the greatest common factor of the variable terms: GCF( x 23 , x 50 ) = x 23 .
Factor out the GCF 16 x 23 from the expression: 16 x 23 − 64 x 50 = 16 x 23 ( 1 − 4 x 27 ) .
The final factorised expression is: 16 x 23 ( 1 − 4 x 27 ) .
Explanation
Understanding the problem We are asked to factorise the expression 16 x 23 − 64 x 50 . This means we want to rewrite the expression as a product of simpler expressions.
Finding the GCF of the coefficients First, let's find the greatest common factor (GCF) of the coefficients, 16 and 64. The GCF of 16 and 64 is 16, since 64 is a multiple of 16 ( 64 = 16 × 4 ).
Finding the GCF of the variable terms Next, let's find the GCF of the variable terms, x 23 and x 50 . The GCF is the lowest power of x that appears in both terms, which is x 23 .
Factoring out the GCF Now, we factor out the GCF, which is 16 x 23 , from the expression:
16 x 23 − 64 x 50 = 16 x 23 ( 1 − 4 x 27 )
Checking for further factorization We check if the expression inside the parenthesis, 1 − 4 x 27 , can be further factorized. In this case, it cannot be factorized further using simple techniques.
Final factorisation Therefore, the fully factorised expression is 16 x 23 ( 1 − 4 x 27 ) .
Examples
Factoring expressions is like simplifying a recipe. Imagine you have a recipe that calls for 16 x 23 cups of flour and 64 x 50 cups of sugar, where x represents the size of a standard measuring cup. By factoring out 16 x 23 , you rewrite the recipe as 16 x 23 ( 1 − 4 x 27 ) . This tells you that you can make 16 batches of a smaller recipe that uses x 23 cups of the mixture ( 1 − 4 x 27 ) . This is useful for scaling recipes or understanding the basic proportions.
To factor the expression 16 x 23 − 64 x 50 , we first find the greatest common factor, which is 16 x 23 . Factoring this out gives us 16 x 23 ( 1 − 4 x 27 ) , which is the final form. This shows how to simplify the expression into a product of simpler terms.
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