Q. \( x^{5} y^{3} (-2 x^{3} y) \).

Q: How do I multiply the monomials in x^5 y^3(-2x^3 y)?

Multiply coefficients and add exponents of like bases: 1\cdot(-2)=-2, x^{5+3}=x^8, y^{3+1}=y^4. Result: -2x^8y^4..

Q: Why do we add exponents when multiplying x^5 and x^3?.

Because of the law of exponents: x^a \cdot x^b = x^{a+b}. You multiply powers by keeping the base and adding the exponents.

Q: What if one factor had no visible coefficient, like x^5?.

An implicit coefficient 1 is assumed. So x^5 is 1\cdot x^5; multiply 1 by the other coefficient.

Answer

Simplify the expression.

\(x^5y^3(-2x^3y)\)

Multiply the coefficients first. The coefficient is \(-2\).

Now use the product rule for exponents. When multiplying powers with the same base, add the exponents.

\(x^5\cdot x^3=x^{5+3}=x^8\)

\(y^3\cdot y=y^{3+1}=y^4\)

So, the simplified expression is:

\(-2x^8y^4\)

Detailed Explanation

Given expression

\(x^{5}y^{3}\bigl(-2x^{3}y\bigr)\)

Separate the numerical coefficients and variable parts.The first factor has coefficient 1 and the second has coefficient −2, so the coefficient part is
\(1\cdot(-2) = -2\)
Multiply the x-powers using the product rule for exponents: a^m · a^n = a^{m+n}.\(x^{5}\cdot x^{3} = x^{5+3} = x^{8}\)
Multiply the y-powers. Treat y as y^{1} and add exponents.\(y^{3}\cdot y^{1} = y^{3+1} = y^{4}\)
Combine the results.Coefficient times variable parts gives
\(-2\cdot x^{8}\cdot y^{4} = -2x^{8}y^{4}\)

Final simplified result:

\(-2x^{8}y^{4}\)

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Algebra FAQs

How do I multiply the monomials in \(x^5 y^3(-2x^3 y)\)?

Multiply coefficients and add exponents of like bases: \(1\cdot(-2)=-2\), \(x^{5+3}=x^8\), \(y^{3+1}=y^4\). Result: \(-2x^8y^4\)..

Why do we add exponents when multiplying \(x^5\) and \(x^3\)?.

Because of the law of exponents: \(x^a \cdot x^b = x^{a+b}\). You multiply powers by keeping the base and adding the exponents.

How do I handle the negative sign in \( -2x^3y\)?

Treat the negative as the coefficient: multiply it with the other coefficient \(1\). So the product picks up the negative: final coefficient \(-2\)..

What order should the final answer be written in?

Standard form places the numeric coefficient first, then variables in alphabetical order with descending exponents: \(-2x^8y^4\) fits this convention.

What if one factor had no visible coefficient, like \(x^5\)?.

An implicit coefficient 1 is assumed. So \(x^5\) is \(1\cdot x^5\); multiply 1 by the other coefficient.

How would zero or negative exponents change the process?

Is distribution needed here or just exponent rules?

No distribution is needed: this is multiplication of monomials, so use coefficient multiplication and exponent addition. Distribution applies when multiplying a term across a sum.

How can I check my answer quickly?

Compare degrees: original x-degrees 5 and 3 give 8, y-degrees 3 and 1 give 4, and sign from coefficient is negative. You can also plug a simple value (e.g., x=1, y=1) into both expressions to confirm equality.

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