Let's simplify the given expression [tex]\(\left(16 t^3\right)^{\frac{3}{2}}\)[/tex] step-by-step to find the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] where it is expressed in the form [tex]\(a t^{\frac{b}{c}}\)[/tex]:
1. Expression: [tex]\(\left(16 t^3\right)^{\frac{3}{2}}\)[/tex]
2. Distribute the exponent [tex]\(\frac{3}{2}\)[/tex] to both parts separately:
[tex]\[
\left(16 t^3\right)^{\frac{3}{2}} = (16)^{\frac{3}{2}} \cdot (t^3)^{\frac{3}{2}}
\][/tex]
3. Simplify [tex]\(16^{\frac{3}{2}}\)[/tex]:
- [tex]\(16\)[/tex] can be written as [tex]\(2^4\)[/tex].
- Therefore, [tex]\((2^4)^{\frac{3}{2}}\)[/tex] simplifies to [tex]\(2^{4 \cdot \frac{3}{2}} = 2^6 = 64\)[/tex].
- So, [tex]\(16^{\frac{3}{2}} = 64\)[/tex].
4. Simplify [tex]\((t^3)^{\frac{3}{2}}\)[/tex]:
- Use the power of a power rule: [tex]\((t^3)^{\frac{3}{2}} = t^{3 \cdot \frac{3}{2}} = t^{\frac{9}{2}}\)[/tex].
- So, [tex]\((t^3)^{\frac{3}{2}} = t^{\frac{9}{2}}\)[/tex].
5. Combine the simplified parts:
[tex]\[
\left(16 t^3\right)^{\frac{3}{2}} = 64 \cdot t^{\frac{9}{2}}
\][/tex]
Now we have the expression in the form [tex]\(a t^{\frac{b}{c}}\)[/tex]:
- [tex]\(a = 64\)[/tex]
- [tex]\(b = 9\)[/tex]
- [tex]\(c = 2\)[/tex]
Thus, the values are:
[tex]\[
\begin{array}{l}
a = 64 \\
b = 9 \\
c = 2
\end{array}
\][/tex]
