((k^2+6k+9)/(k^2+12k+27))×(k^2+9)/((k^2+12k+27))
express in lowest terms
k ^ 2 + 6 k + 9 = ( x + 3 ) * ( x + 3 )
k ^ 2 + 12 k + 27 = ( x + 9 ) * ( x + 3 )
((k^2+6k+9)/(k+9)*(k+3)))*(k^2+9)/((k^2+12k+27))=
((k+3)*(k+3))/((x+9)*(k+3)) * ((k^2+9)/(((x+9)*(k+3))=
((k+3))/((x+9)) * ((k^2+9)/(((x+9)*(k+3))=
(k^2+9)/((x+9)*(k+9))=
(k^2+9)/(x+9)^2
(k^2+9)/((k+9)*(k+9))=
(k^2+9)/(k+9)^
To express the given expression in its lowest terms, we need to simplify it by canceling out any common factors in the numerator and denominator.
Given expression: ((k^2 + 6k + 9) / (k^2 + 12k + 27)) × (k^2 + 9) / (k^2 + 12k + 27)
First, let's factorize the quadratic expressions in both the numerator and denominator.
k^2 + 6k + 9 can be factored as (k + 3) × (k + 3), since (k + 3) × (k + 3) = k^2 + 6k + 9.
k^2 + 12k + 27 can be factored as (k + 3) × (k + 9), since (k + 3) × (k + 9) = k^2 + 12k + 27.
Now, we can rewrite the expression, factoring the common terms in the numerator and denominator:
((k + 3) × (k + 3) / (k + 3) × (k + 9)) × (k^2 + 9) / (k^2 + 12k + 27)
Next, cancel out the common factors (k + 3) in the numerator and denominator:
(k + 3) / (k + 9) × (k^2 + 9) / (k^2 + 12k + 27)
The simplified expression, in its lowest terms, is:
(k + 3)(k^2 + 9) / (k + 9)(k^2 + 12k + 27)