Given:
[tex]$\frac{\left(\frac{(4 r)^{3}}{15 t^{4}}\right)}{\left(\frac{16 r}{(3 t)^{2}}\right)}[/tex]
To find:
The simplified fraction.
Solution:
Step 1: Simplify the numerator
[tex]$\frac{(4 r)^{3}}{15 t^{4}}=\frac{4^3 r^{3}}{15 t^{4}}=\frac{64 r^{3}}{15 t^{4}}[/tex]
Step 2: Simplify the denominator
[tex]$\frac{16 r}{(3 t)^{2}}=\frac{16 r}{3^2 t^{2}}= \frac{16 r}{9 t^{2}}[/tex]
Step 3: Using step 1 and step 2
[tex]$\frac{\left(\frac{(4 r)^{3}}{15 t^{4}}\right)}{\left(\frac{16 r}{(3 t)^{2}}\right)}=\frac{\left(\frac{64 r^{3}}{15 t^{4}}\right)}{\left(\frac{16 r}{9 t^{2}} \right)}[/tex]
Step 4: Using fraction rule:
[tex]$\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{a \cdot d}{b \cdot c}[/tex]
[tex]$\frac{\left(\frac{64 r^{3}}{15 t^{4}}\right)}{\left(\frac{16 r}{9 t^{2}}\right)}=\frac{64r^3 \cdot 9t^2}{16 r \cdot 15 t^4}[/tex]
Cancel the common factor r and t², we get
[tex]$=\frac{64 r^{2} \cdot 9 }{16 \cdot 15 t^2 }[/tex]
Cancel the common factors 16 and 3 on both numerator and denominator.
[tex]$=\frac{4 r^{2} \cdot 3 }{ 5 t^2 }[/tex]
[tex]$=\frac{12 r^{2} }{ 5 t^2 }[/tex]
[tex]$\frac{\left(\frac{(4 r)^{3}}{15 t^{4}}\right)}{\left(\frac{16 r}{(3 t)^{2}}\right)}=\frac{12 r^{2} }{ 5 t^2 }[/tex]
The simplified fraction is [tex]\frac{12 r^{2} }{ 5 t^2 }[/tex].
