Let's solve the equation step-by-step: [tex]\(\frac{t \cdot t^3}{\sqrt{t}} = t^x\)[/tex].
1. Understanding the left-hand side: We start with the left-hand side:
[tex]\[
\frac{t \cdot t^3}{\sqrt{t}}
\][/tex]
2. Combine the exponents: Notice that multiplying the terms [tex]\(t\)[/tex] and [tex]\(t^3\)[/tex] in the numerator involves adding their exponents. So,
[tex]\[
t \cdot t^3 = t^{1+3} = t^4
\][/tex]
3. Simplify the fraction: Now the expression becomes:
[tex]\[
\frac{t^4}{\sqrt{t}}
\][/tex]
4. Expressing the square root in exponent form: Recall that the square root of [tex]\(t\)[/tex] can be written as:
[tex]\[
\sqrt{t} = t^{1/2}
\][/tex]
5. Simplifying further by combining the exponents: Using the property of exponents for division [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex]:
[tex]\[
\frac{t^4}{t^{1/2}} = t^{4 - 1/2}
\][/tex]
6. Subtract the exponents:
[tex]\[
4 - \frac{1}{2} = \frac{8}{2} - \frac{1}{2} = \frac{7}{2}
\][/tex]
So, the left-hand side simplifies to:
[tex]\[
t^{7/2}
\][/tex]
7. Equate with the right-hand side: Now, our equation looks like:
[tex]\[
t^{7/2} = t^x
\][/tex]
8. Comparing exponents: Since the bases [tex]\(t\)[/tex] are the same, the exponents must be equal. Therefore, we equate the exponents:
[tex]\[
\frac{7}{2} = x
\][/tex]
Thus, the solution is:
[tex]\[ x = \frac{7}{2} \][/tex]
So, the simplified left-hand side expression is [tex]\(t^{7/2}\)[/tex], and the value of [tex]\(x\)[/tex] that satisfies the equation is [tex]\( \frac{7}{2} \)[/tex].
