To simplify the given expression [tex]\(\frac{5 t^6}{9 t^{17}}\)[/tex] using only positive exponents, we need to follow these steps:
1. Identify the common base term [tex]\(t\)[/tex]:
The base [tex]\(t\)[/tex] appears in both the numerator and the denominator.
2. Subtract the exponents of the common base term in the numerator and the denominator:
According to the laws of exponents, when you divide terms with the same base, you subtract the exponents:
[tex]\[
t^6 \div t^{17} = t^{6 - 17} = t^{-11}
\][/tex]
3. Rewrite the expression using the simplified exponent:
So, [tex]\(\frac{t^6}{t^{17}} = t^{-11}\)[/tex].
4. Substitute the simplified base term back into the fraction:
The fractional expression now becomes:
[tex]\[
\frac{5 t^{-11}}{9}
\][/tex]
5. Express the term with negative exponent using positive exponents:
Recall that [tex]\(t^{-11} = \frac{1}{t^{11}}\)[/tex]. So, we can rewrite the expression as:
[tex]\[
\frac{5}{9} \times \frac{1}{t^{11}} = \frac{5}{9 t^{11}}
\][/tex]
Thus, the simplified expression using only positive exponents is:
[tex]\[
\boxed{\frac{5}{9 t^{11}}}
\][/tex]
