Answer :
To simplify the expression [tex]\(5vy^3 \cdot 2x^7x^4v^{-9} \cdot 5y^{-3}\)[/tex] using only positive exponents, follow these steps:
1. Combine the Constants:
- The constants in the expression are 5, 2, and 5. Multiply them together:
[tex]\[
5 \times 2 \times 5 = 50
\][/tex]
2. Combine the Variables:
- For [tex]\(v\)[/tex]:
- The exponents for [tex]\(v\)[/tex] are 1 from [tex]\(v\)[/tex] and -9 from [tex]\(v^{-9}\)[/tex].
- Add the exponents: [tex]\(1 + (-9) = -8\)[/tex].
- For [tex]\(x\)[/tex]:
- The exponents for [tex]\(x\)[/tex] are 7 and 4 from [tex]\(x^7\)[/tex] and [tex]\(x^4\)[/tex].
- Add the exponents: [tex]\(7 + 4 = 11\)[/tex].
- For [tex]\(y\)[/tex]:
- The exponents for [tex]\(y\)[/tex] are 3 from [tex]\(y^3\)[/tex] and -3 from [tex]\(y^{-3}\)[/tex].
- Add the exponents: [tex]\(3 + (-3) = 0\)[/tex], which means [tex]\(y^0 = 1\)[/tex].
3. Simplify the Expression:
- Given the result for [tex]\(y^0\)[/tex], the [tex]\(y\)[/tex] terms cancel each other out.
- The expression is simplified to:
[tex]\[
50 \cdot x^{11} \cdot v^{-8}
\][/tex]
4. Convert to Positive Exponents:
- Rewrite [tex]\(v^{-8}\)[/tex] using a positive exponent:
[tex]\[
v^{-8} = \frac{1}{v^8}
\][/tex]
- The final simplified expression is:
[tex]\[
\frac{50 \cdot x^{11}}{v^8}
\][/tex]
So, the simplified expression with positive exponents is [tex]\(\frac{50 \cdot x^{11}}{v^8}\)[/tex].
1. Combine the Constants:
- The constants in the expression are 5, 2, and 5. Multiply them together:
[tex]\[
5 \times 2 \times 5 = 50
\][/tex]
2. Combine the Variables:
- For [tex]\(v\)[/tex]:
- The exponents for [tex]\(v\)[/tex] are 1 from [tex]\(v\)[/tex] and -9 from [tex]\(v^{-9}\)[/tex].
- Add the exponents: [tex]\(1 + (-9) = -8\)[/tex].
- For [tex]\(x\)[/tex]:
- The exponents for [tex]\(x\)[/tex] are 7 and 4 from [tex]\(x^7\)[/tex] and [tex]\(x^4\)[/tex].
- Add the exponents: [tex]\(7 + 4 = 11\)[/tex].
- For [tex]\(y\)[/tex]:
- The exponents for [tex]\(y\)[/tex] are 3 from [tex]\(y^3\)[/tex] and -3 from [tex]\(y^{-3}\)[/tex].
- Add the exponents: [tex]\(3 + (-3) = 0\)[/tex], which means [tex]\(y^0 = 1\)[/tex].
3. Simplify the Expression:
- Given the result for [tex]\(y^0\)[/tex], the [tex]\(y\)[/tex] terms cancel each other out.
- The expression is simplified to:
[tex]\[
50 \cdot x^{11} \cdot v^{-8}
\][/tex]
4. Convert to Positive Exponents:
- Rewrite [tex]\(v^{-8}\)[/tex] using a positive exponent:
[tex]\[
v^{-8} = \frac{1}{v^8}
\][/tex]
- The final simplified expression is:
[tex]\[
\frac{50 \cdot x^{11}}{v^8}
\][/tex]
So, the simplified expression with positive exponents is [tex]\(\frac{50 \cdot x^{11}}{v^8}\)[/tex].
