Alright, let's break down the detailed, step-by-step solution for the expression [tex]\( 3\left(5 t^{\frac{3}{8}}-2 t^{\frac{-2}{7}}\right) \)[/tex]:
1. Initial Expression:
We start with the expression:
[tex]\[
3\left(5 t^{\frac{3}{8}} - 2 t^{\frac{-2}{7}}\right)
\][/tex]
2. Distribute the Constant:
Next, we distribute the constant 3 across the terms inside the parentheses:
[tex]\[
3 \cdot (5 t^{\frac{3}{8}}) - 3 \cdot (2 t^{\frac{-2}{7}})
\][/tex]
Simplifying each term, we get:
[tex]\[
15 t^{\frac{3}{8}} - 6 t^{\frac{-2}{7}}
\][/tex]
3. Combine the Exponential Terms:
Notice that the exponent terms have different bases. To simplify, we can represent these powers explicitly:
[tex]\[
15 t^{\frac{3}{8}} - 6 t^{-\frac{2}{7}}
\][/tex]
4. Rewrite the Negative Exponent:
The term [tex]\( 6 t^{-\frac{2}{7}} \)[/tex] with a negative exponent can be rewritten as a fraction:
[tex]\[
6 t^{-\frac{2}{7}} = \frac{6}{t^{\frac{2}{7}}}
\][/tex]
5. Common Denominator:
To combine the terms into a single fraction with a common denominator, we need to match the base powers. First, note the term [tex]\( 15 t^{\frac{3}{8}} \)[/tex] can be rewritten using a common base:
[tex]\[
15 t^{\frac{3}{8}} = \frac{15 t^{\frac{3}{8}} \cdot t^{\frac{2}{7}}}{t^{\frac{2}{7}}} = \frac{15 t^{\frac{3}{8} + \frac{2}{7}}}{t^{\frac{2}{7}}}
\][/tex]
6. Simplify the Exponents:
Adding the exponents in the numerator:
[tex]\[
\frac{3}{8} + \frac{2}{7} = \frac{21}{56} + \frac{16}{56} = \frac{37}{56}
\][/tex]
So we have:
[tex]\[
15 t^{\frac{37}{56}} \cdot \frac{1}{t^{\frac{2}{7}}} = \frac{15 t^{\frac{37}{56}}}{t^{\frac{2}{7}}}
\][/tex]
7. Simplify the Common Denominator:
Recognizing [tex]\( \frac{2}{7} \)[/tex] simplifies directly to our original term form:
[tex]\[
\frac{15 t^{\frac{37}{56}} - 6}{t^{\frac{2}{7}}}
\][/tex]
8. Combine the Fractions and Operations:
Now simplify [tex]\( t^{\frac{2}{7}} \)[/tex]. Realizing a division by reaffirming original terms:
[tex]\[
15 t^{\frac{0.660714285714286}} - 6 \text{ explicitly simplified is }
\][/tex]
Thus, our final combined form:
[tex]\[
\frac{3(5 t^{0.660714285714286} - 2)}{t^{0.285714285714286}}
\][/tex]
So our final result for simplifying the expression [tex]\(3\left(5 t^{\frac{3}{8}}-2 t^{\frac{-2}{7}}\right)\)[/tex] is:
[tex]\[
\boxed{3\left(5 t^{0.660714285714286} - 2\right)t^{-0.285714285714286}}
\][/tex]
