Discover answers to your questions with Westonci.ca, the leading Q&A platform that connects you with knowledgeable experts. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
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]
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]
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
