Answer:
[tex]\textsf{55)} \quad \textsf{D)}\;\;\dfrac{m^{\frac{7}{12}}n^{\frac12}}{n^2}[/tex]
[tex]\textsf{56)} \quad \textsf{B)}\;\;\dfrac{x^{\frac{3}{8}}}{y^{4}x^{3}}[/tex]
Step-by-step explanation:
Given expression:
[tex]\dfrac{m^{\frac{5}{4}}m^{2}}{\left(m^{-\frac{4}{3}}n^{-\frac{3}{4}}\right)^{-2}}[/tex]
Begin by applying the product rule of exponents to the numerator:
[tex]\dfrac{m^{\frac{5}{4}+2}}{\left(m^{-\frac{4}{3}}n^{-\frac{3}{4}}\right)^{-2}} \\\\\\\\ \dfrac{m^{\frac{13}{4}}}{\left(m^{-\frac{4}{3}}n^{-\frac{3}{4}}\right)^{-2}} \\\\\\\\[/tex]
Now, apply the power of a product rule to the denominator:
[tex]\dfrac{m^{\frac{13}{4}}}{\left(m^{-\frac{4}{3}}\right)^{-2}\left(n^{-\frac{3}{4}}\right)^{-2}}[/tex]
Apply the power of a power rule to simplify the denominator:
[tex]\dfrac{m^{\frac{13}{4}}}{m^{-\frac{4}{3}\cdot (-2)} n^{-\frac{3}{4}\cdot (-2)}} \\\\\\ \dfrac{m^{\frac{13}{4}}}{m^{\frac{8}{3}}n^{\frac64}} \\\\\\ \dfrac{m^{\frac{13}{4}}}{m^{\frac{8}{3}}n^{\frac32}}[/tex]
Now, apply the quotient rule to the m-variable:
[tex]\dfrac{m^{\frac{13}{4}-\frac{8}{3}}}{n^{\frac32}} \\\\\\ \dfrac{m^{\frac{39}{12}-\frac{32}{12}}}{n^{\frac32}} \\\\\\ \dfrac{m^{\frac{7}{12}}}{n^{\frac32}}[/tex]
Rewrite the exponent of the n-variable in the denominator as 2 - 1/2:
[tex]\dfrac{m^{\frac{7}{12}}}{n^{2-\frac12}}[/tex]
Apply the product rule:
[tex]\dfrac{m^{\frac{7}{12}}}{n^2\cdot n^{-\frac12}}[/tex]
Finally, apply the negative exponent rule:
[tex]\dfrac{m^{\frac{7}{12}}\cdot n^{-\left(-\frac12\right)}}{n^2} \\\\\\ \dfrac{m^{\frac{7}{12}}n^{\frac12}}{n^2}[/tex]
Therefore, the simplified expression where all exponents are positive and there are no fractional exponents in denominator is:
[tex]\LARGE\boxed{\boxed{\dfrac{m^{\frac{7}{12}}n^{\frac12}}{n^2}}}[/tex]
[tex]\dotfill[/tex]
Given expression:
[tex]\left(\dfrac{\left(x^{\frac13}y^{\frac13}\right)^{\frac74}}{xy^{\frac54}\cdot x^{\frac43}y^2}\right)^{\frac32}[/tex]
Begin by applying the power of a product rule and power of a power rule to the numerator:
[tex]\left(\dfrac{\left(x^{\frac13}\right)^{\frac74}\left(y^{\frac13}\right)^{\frac74}}{xy^{\frac54}\cdot x^{\frac43}y^2}\right)^{\frac32} \\\\\\\\ \left(\dfrac{x^{\frac13\cdot \frac74}\cdot y^{\frac13\cdot \frac74}}{xy^{\frac54}\cdot x^{\frac43}y^2}\right)^{\frac32} \\\\\\\\ \left(\dfrac{x^{\frac{7}{12}}\cdot y^{\frac{7}{12}}}{xy^{\frac54}\cdot x^{\frac43}y^2}\right)^{\frac32}[/tex]
Rearrange the terms in the denominator to collect like variables:
[tex]\left(\dfrac{x^{\frac{7}{12}}\cdot y^{\frac{7}{12}}}{x\cdot x^{\frac43} \cdot y^2\cdot y^{\frac54}}\right)^{\frac32}[/tex]
Apply the product rule to the denominator:
[tex]\left(\dfrac{x^{\frac{7}{12}}\cdot y^{\frac{7}{12}}}{x^{1+\frac43} \cdot y^{2+\frac54}}\right)^{\frac32} \\\\\\\\ \left(\dfrac{x^{\frac{7}{12}}\cdot y^{\frac{7}{12}}}{x^{\frac73} \cdot y^{\frac{13}{4}}}\right)^{\frac32}[/tex]
Now, apply the quotient rule:
[tex]\left(x^{\frac{7}{12}-\frac73}\cdot y^{\frac{7}{12}-\frac{13}{4}}\right)^{\frac32} \\\\\\ \left(x^{-\frac74}\cdot y^{-\frac{8}{3}}\right)^{\frac32}[/tex]
Apply the power of a product rule and power of a power rule:
[tex]\left(x^{-\frac74}\right)^{\frac32} \cdot \left(y^{-\frac{8}{3}}\right)^{\frac32} \\\\\\ x^{-\frac74}\cdot \frac32} \cdot y^{-\frac{8}{3}\cdot \frac32} \\\\\\ x^{-\frac{21}{8}} \cdot y^{-4}[/tex]
Apply the negative exponent rule to the y-variable:
[tex]\dfrac{x^{-\frac{21}{8}}}{y^{4}}[/tex]
Rewrite the exponent of x as 3/8 - 3:
[tex]\dfrac{x^{\frac{3}{8}-3}}{y^{4}}[/tex]
Apply the product rule:
[tex]\dfrac{x^{\frac{3}{8}}\cdot x^{-3}}{y^{4}}[/tex]
Finally, apply the negative exponent rule to x⁻³:
[tex]\dfrac{x^{\frac{3}{8}}}{y^{4}x^3}[/tex]
Therefore, the simplified expression where all exponents are positive and there are no fractional exponents in denominator is:
[tex]\LARGE\boxed{\boxed{\dfrac{x^{\frac{3}{8}}}{y^{4}x^3}}}[/tex]
[tex]\dotfill[/tex]
[tex]\boxed{\begin{array}{c}\underline{\textsf{Exponent Rules}}\\\\\textsf{Product:}\;\;a^m \times a^n=a^{m+n}\\\\\textsf{Quotient:}\;\;a^m \div a^n=a^{m-n}\\\\\textsf{Power of a Power:}\;\;(a^m)^n=a^{mn}\\\\\textsf{Power of a Product:}\;\;(ab)^m=a^mb^m\\\\\textsf{Negative Exponent:}\;\;a^{-m}=\dfrac{1}{a^m}\end{array}}[/tex]