Answer :
To solve the given exponential expression:
[tex]\[ \left(-5 x^4 y^{-5}\right)\left(3 x^{-1} y\right) \][/tex]
we will break down the multiplication into manageable parts, combining the coefficients and exponents separately.
### Step 1: Multiply the coefficients
The coefficients are \(-5\) and \(3\):
[tex]\[ -5 \cdot 3 = -15 \][/tex]
### Step 2: Combine the exponents of \(x\)
The exponents of \(x\) in the two terms are \(4\) and \(-1\):
[tex]\[ x^4 \cdot x^{-1} = x^{4 + (-1)} = x^3 \][/tex]
### Step 3: Combine the exponents of \(y\)
The exponents of \(y\) in the two terms are \(-5\) and \(1\):
[tex]\[ y^{-5} \cdot y^1 = y^{-5 + 1} = y^{-4} \][/tex]
### Step 4: Combine the results
Combining the coefficient and the combined exponents, the simplified expression is:
[tex]\[ -15 x^3 y^{-4} \][/tex]
### Step 5: Write the expression in a simplified form
Since \(y^{-4}\) is the same as \(\frac{1}{y^4}\), we can rewrite the expression as:
[tex]\[ -15 x^3 y^{-4} = \frac{-15 x^3}{y^4} \][/tex]
Thus, the simplified form of \(\left(-5 x^4 y^{-5}\right)\left(3 x^{-1} y\right)\) is:
[tex]\[ \frac{-15 x^3}{y^4} \][/tex]
### Conclusion
Given the multiple-choice options:
A) \(\frac{-15 x^3}{y^4}\)
B) \(\frac{-15 x^5}{y^6}\)
C) \(\frac{-2 x^3}{y^4}\)
D) \(-15 x^3 y^6\)
The correct answer is:
A) [tex]\(\frac{-15 x^3}{y^4}\)[/tex]
[tex]\[ \left(-5 x^4 y^{-5}\right)\left(3 x^{-1} y\right) \][/tex]
we will break down the multiplication into manageable parts, combining the coefficients and exponents separately.
### Step 1: Multiply the coefficients
The coefficients are \(-5\) and \(3\):
[tex]\[ -5 \cdot 3 = -15 \][/tex]
### Step 2: Combine the exponents of \(x\)
The exponents of \(x\) in the two terms are \(4\) and \(-1\):
[tex]\[ x^4 \cdot x^{-1} = x^{4 + (-1)} = x^3 \][/tex]
### Step 3: Combine the exponents of \(y\)
The exponents of \(y\) in the two terms are \(-5\) and \(1\):
[tex]\[ y^{-5} \cdot y^1 = y^{-5 + 1} = y^{-4} \][/tex]
### Step 4: Combine the results
Combining the coefficient and the combined exponents, the simplified expression is:
[tex]\[ -15 x^3 y^{-4} \][/tex]
### Step 5: Write the expression in a simplified form
Since \(y^{-4}\) is the same as \(\frac{1}{y^4}\), we can rewrite the expression as:
[tex]\[ -15 x^3 y^{-4} = \frac{-15 x^3}{y^4} \][/tex]
Thus, the simplified form of \(\left(-5 x^4 y^{-5}\right)\left(3 x^{-1} y\right)\) is:
[tex]\[ \frac{-15 x^3}{y^4} \][/tex]
### Conclusion
Given the multiple-choice options:
A) \(\frac{-15 x^3}{y^4}\)
B) \(\frac{-15 x^5}{y^6}\)
C) \(\frac{-2 x^3}{y^4}\)
D) \(-15 x^3 y^6\)
The correct answer is:
A) [tex]\(\frac{-15 x^3}{y^4}\)[/tex]
