Question

if ( t^7 = 4m ) and ( t^9 = m^3 ), then what is the value of ( t ) in terms of ( m )?

(a) ( 2sqrt{m} )

(b) ( 2m )

(c) ( \frac{m}{2} )

(d) ( \frac{msqrt{2}}{4} )

(e) ( msqrt{2} )

Explanation:

Step1: Isolate $t^7$ and $t^9$

We know:
$t^7 = 4m$ --- (1)
$t^9 = m^3$ --- (2)

Step2: Express $t^9$ as $t^7 \cdot t^2$

Substitute $t^9$ in (2) using (1):
$t^7 \cdot t^2 = m^3$
$4m \cdot t^2 = m^3$

Step3: Solve for $t^2$

Divide both sides by $4m$ ($m
eq 0$):
$t^2 = \frac{m^3}{4m} = \frac{m^2}{4}$

Step4: Solve for $t$

Take square root of both sides:
$t = \sqrt{\frac{m^2}{4}} = \frac{m}{2}$

Answer:

(C) $\frac{m}{2}$