Sure! Let's rewrite the rational expression \(\frac{6t + 7}{30t + 35}\) with the denominator \(5s(6t + 7)\).
1. Factor the denominator of the original expression:
The given denominator is \( 30t + 35 \).
We can factor out a 5 from both terms:
[tex]\[
30t + 35 = 5(6t + 7)
\][/tex]
2. Rewrite the original rational expression with the factored denominator:
Expressing the original rational expression:
[tex]\[
\frac{6t + 7}{30t + 35} = \frac{6t + 7}{5(6t + 7)}
\][/tex]
3. Match the new denominator:
We need the new equivalent denominator to be \( 5s(6t + 7) \).
Observe that the new denominator is obtained by multiplying \( 5(6t + 7) \) by \( s \):
[tex]\[
5(6t + 7) \rightarrow 5s(6t + 7)
\][/tex]
4. Adjust the numerator to maintain the equivalence:
To keep the expression equivalent when adjusting the denominator, we must multiply the numerator by \( s \):
[tex]\[
\frac{6t + 7}{5(6t + 7)} = \frac{s \cdot (6t + 7)}{5s(6t + 7)}
\][/tex]
So, the rewritten rational expression with the new denominator is:
[tex]\[
\frac{6t + 7}{30t + 35} = \frac{s(6t + 7)}{5s(6t + 7)}
\][/tex]
Thus, the resulting step-by-step process shows that:
[tex]\[
\frac{6t + 7}{30t + 35} = \frac{s(6t + 7)}{5s(6t + 7)}
\][/tex]
And that completes the solution!
