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Simplify the exponential expression. Assume that variables represent nonzero real numbers.

[tex]\[
\left(-5 x^4 y^{-5}\right)\left(3 x^{-1} y\right)
\][/tex]

A) \(\frac{-15 x^3}{y^4}\)

B) \(\frac{-15 x^5}{y^6}\)

C) \(\frac{-2 x^3}{y^4}\)

D) [tex]\(-15 x^3 y^6\)[/tex]


Sagot :

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]