The quotient is:
[tex]\frac{1}{(t+4)^2} \text{ or } \frac{1}{t^2+8t+16}[/tex]
Solution:
Given expression is:
[tex]\frac{t+3}{\frac{t+4}{t^{2}+7t+12}}[/tex]
We have to find the quotient
From given,
[tex]\frac{t+3}{\frac{t+4}{t^{2}+7t+12}} = \frac{t+3}{t+4} \times \frac{1}{(t^2+7t+12)} ------ eqn 1[/tex]
[tex]Let\ us\ factor\ (t^2+7t+12)[/tex]
Split the middle term
[tex]t^2+7t+12 = t^2 + 3t+4t+12[/tex]
Group the terms
[tex](t^2+3t) + (4t+12)[/tex]
Take the common factor out
[tex]t(t+3)+4(t+3)[/tex]
Again take (t+3) as common factor
[tex](t+3)(t+4)[/tex]
Substitute in eqn 1
[tex]\frac{t+3}{\frac{t+4}{t^{2}+7t+12}} = \frac{t+3}{t+4} \times \frac{1}{(t+3)(t+4)}[/tex]
Cancel the common factors
[tex]\frac{t+3}{\frac{t+4}{t^{2}+7t+12}} = \frac{1}{(t+4)^2}[/tex]
Therefore, quotient is:
[tex]\frac{t+3}{\frac{t+4}{t^{2}+7t+12}} = \frac{1}{(t+4)^2} = \frac{1}{t^2+8t+16}[/tex]
The quotient of the given expression is [tex]\rm 1/(t+4)^2[/tex] and this can be determined by using the arithmetic operations.
Given :
Expression -- [tex]\rm \dfrac{t+3}{\dfrac{t+4}{t^2+7t+12}}[/tex]
The following steps can be used in order to evaluate the given expression:
Step 1 - The arithmetic operations can be used in order to evaluate the given expression.
Step 2 - Write the given expression.
[tex]\rm \dfrac{t+3}{\dfrac{t+4}{t^2+7t+12}}[/tex]
Step 3 - Now, factorize the above expression.
[tex]\rm \dfrac{t+3}{\dfrac{t+4}{t^2+4t+3t+12}}[/tex]
[tex]\rm \dfrac{t+3}{\dfrac{t+4}{t(t+4)+3(t+4)}}[/tex]
[tex]\rm \dfrac{t+3}{\dfrac{t+4}{(t+4)(t+3)}}[/tex]
Step 4 - Simplify the above expression.
[tex]\rm \dfrac{1}{(t + 4)^2}[/tex]
For more information, refer to the link given below:
https://brainly.com/question/6810544
