Answer:
[tex]\textsf{(a)}\quad \dfrac{1}{8}x[/tex]
[tex]\textsf{(b)}\quad \dfrac{1}{256}x^{\frac32}[/tex]
[tex]\textsf{(c)}\quad 16x^{-2}[/tex]
Step-by-step explanation:
To rewrite the given expressions in the form of [tex]kx^n[/tex], where k and n are simplified constants, we can use the rules of exponents.
[tex]\hrulefill[/tex]
Part (a)
[tex]\textsf{Given that\;\;$a=\dfrac{1}{64}x^2$, then:}[/tex]
[tex]a^{\frac{1}{2}}=\left(\dfrac{1}{64}x^2\right)^{\frac{1}{2}}[/tex]
[tex]\textsf{Apply the power of a product exponent rule:} \quad (ab)^c=a^{c}b^c[/tex]
[tex]=\left(\dfrac{1}{64}\right)^{\frac{1}{2}}(x^2)^{\frac{1}{2}}[/tex]
[tex]\textsf{Apply the power of a quotient exponent rule:} \quad \left(\dfrac{a}{b}\right)^c=\dfrac{a^c}{b^c}[/tex]
[tex]=\left(\dfrac{1^{\frac{1}{2}}}{64^{\frac{1}{2}}}\right)(x^2)^{\frac{1}{2}}\\\\\\=\dfrac{1}{8}(x^2)^{\frac12}[/tex]
[tex]\textsf{Apply the power of a power exponent rule:} \quad (a^b)^c=a^{bc}[/tex]
[tex]=\dfrac{1}{8}x^{(2 \times \frac12)}\\\\\\=\dfrac{1}{8}x^{1}\\\\\\=\dfrac{1}{8}x[/tex]
[tex]\hrulefill[/tex]
Part (b)
[tex]\textsf{Given that\;\;$b=\dfrac{16}{\sqrt{x}}$,\;then:}[/tex]
[tex]\dfrac{16}{b^3}=\dfrac{16}{\left(\dfrac{16}{\sqrt{x}}\right)^3}[/tex]
[tex]\textsf{Rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$:}[/tex]
[tex]=\dfrac{16}{\left(\dfrac{16}{x^{\frac12}}\right)^3}[/tex]
[tex]\textsf{Apply the power of a quotient exponent rule:} \quad \left(\dfrac{a}{b}\right)^c=\dfrac{a^c}{b^c}[/tex]
[tex]=\dfrac{16}{\left(\dfrac{16^3}{\left(x^{\frac12}}\right)^3\right)}\\\\\\=\dfrac{16}{\left(\dfrac{4096}{\left(x^{\frac12}}\right)^3\right)}[/tex]
[tex]\textsf{Apply the power of a power exponent rule:} \quad (a^b)^c=a^{bc}[/tex]
[tex]=\dfrac{16}{\left(\dfrac{4096}{x^{\frac32}}\right)}[/tex]
[tex]\textsf{Apply the fraction rule:} \quad \dfrac{a}{\frac{b}{c}}=\dfrac{ac}{b}[/tex]
[tex]=\dfrac{16x^{\frac32}}{4096}\\\\\\=\dfrac{1}{256}x^{\frac32}[/tex]
[tex]\hrulefill[/tex]
Part (c)
[tex]\textsf{Given that\;\;$a=\dfrac{1}{64}x^2$\;\;and\;\;$b=\dfrac{16}{\sqrt{x}}$,\;then:}[/tex]
[tex]\left(\dfrac{ab}{2}\right)^{-\frac{4}{3}}=\left(\dfrac{\left(\dfrac{1}{64}x^2\right)\left(\dfrac{16}{\sqrt{x}}\right)}{2}\right)^{-\frac{4}{3}}[/tex]
Simplify the numerator:
[tex]=\left(\dfrac{\left(\dfrac{16x^2}{64\sqrt{x}}\right)}{2}\right)^{-\frac{4}{3}}\\\\\\\\=\left(\dfrac{\left(\dfrac{x^{\left(2-\frac12\right)}}{4}\right)}{2}\right)^{-\frac{4}{3}}\\\\\\\\=\left(\dfrac{\left(\dfrac{x^{\frac32}}{4}\right)}{2}\right)^{-\frac{4}{3}}[/tex]
[tex]\textsf{Apply the fraction rule:} \quad \dfrac{\frac{a}{b}}{c}=\dfrac{a}{bc}[/tex]
[tex]=\left(\dfrac{x^{\frac32}}{4 (2)}\right)^{-\frac{4}{3}}\\\\\\\\=\left(\dfrac{x^{\frac32}}{8}\right)^{-\frac{4}{3}}[/tex]
[tex]\textsf{Apply the power of a quotient exponent rule:} \quad \left(\dfrac{a}{b}\right)^c=\dfrac{a^c}{b^c}[/tex]
[tex]=\dfrac{\left(x^{\frac32}\right)^{-\frac43}}{(8)^{-\frac43}}\\\\\\\\=\dfrac{\left(x^{\frac32}\right)^{-\frac43}}{\frac{1}{16}}[/tex]
[tex]\textsf{Apply the power of a power exponent rule:} \quad (a^b)^c=a^{bc}[/tex]
[tex]=\dfrac{x^{\left(\frac{3}{2}\left(-\frac{4}{3}\right)}\right)}{\frac{1}{16}}\\\\\\\\=\dfrac{x^{-\frac{12}{6}}}{\frac{1}{16}}\\\\\\\\=\dfrac{x^{-2}}{\frac{1}{16}}[/tex]
[tex]\textsf{Apply the fraction rule:} \quad \dfrac{a}{\frac{b}{c}}=\dfrac{ac}{b}[/tex]
[tex]=\dfrac{16x^{-2}}{1}\\\\\\=16x^{-2}[/tex]