If `a = root(3)(t)` and `b = t^(2)`, then `(b)/(a^(5))=`

Options

A. `t^(-(1)/(3))`
B. `t^((1)/(3))`
C. `t^((5)/(6))`
D. `t^((6)/(5))`\

Explanation

To solve the problem, we need to find the value of \( \frac{b}{a^5} \) given that \( a = \sqrt[3]{t} \) and \( b = t^2 \).

### Step-by-Step Solution:

1. **Identify the values of \( a \) and \( b \)**:
- We have \( a = \sqrt[3]{t} \) and \( b = t^2 \).

2. **Substitute the values of \( a \) and \( b \) into the expression \( \frac{b}{a^5} \)**:
- Substitute \( b \) with \( t^2 \) and \( a \) with \( \sqrt[3]{t} \):
\[ \frac{b}{a^5} = \frac{t^2}{(\sqrt[3]{t})^5} \]

3. **Simplify \( a^5 \)**:
- Recall that \( \sqrt[3]{t} = t^{1/3} \), thus:
\[ a^5 = (t^{1/3})^5 = t^{5/3} \]

4. **Rewrite the expression**:
- Now we can rewrite the expression:
\[ \frac{b}{a^5} = \frac{t^2}{t^{5/3}} \]

5. **Apply the laws of exponents**:
- When dividing like bases, we subtract the exponents:
\[ \frac{t^2}{t^{5/3}} = t^{2 - 5/3} \]

6. **Convert \( 2 \) to a fraction with a common denominator**:
- Convert \( 2 \) to a fraction:
\[ 2 = \frac{6}{3} \]
- Now substitute:
\[ t^{2 - 5/3} = t^{\frac{6}{3} - \frac{5}{3}} = t^{\frac{1}{3}} \]

7. **Final Result**:
- Therefore, the final result is:
\[ \frac{b}{a^5} = t^{\frac{1}{3}} \]