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

Q: What is the fully simplified form of x^{5}y^{3}(-2x^{3}y)? Do not use arrows.

Multiply coefficients and add exponents: -2 \cdot x^{5+3} \cdot y^{3+1} = -2x^{8}y^{4}.

Q: How do you multiply powers with the same base?

Use x^{a}\cdot x^{b}=x^{a+b}. Here x^{5}\cdot x^{3}=x^{8} and y^{3}\cdot y^{1}=y^{4}.

Q: How is the negative sign handled in multiplication?

Multiply the coefficients including sign: 1\cdot(-2)=-2. The negative stays in front of the final product: -2x^{8}y^{4}..

Q: What if exponents are zero, e.g., x^{0}y^{3}(-2x^{3}y)?.

x^{0}=1, so the product becomes -2\cdot x^{3}\cdot y^{4} = -2x^{3}y^{4}..

Q: Are the exponent rules valid for non-integer exponents?

Yes: x^{a}\cdot x^{b}=x^{a+b} holds for real a,b where each expression is defined (watch bases and fractional exponents requiring positive bases for even denominators).

Q: What happens if x or y equals 0?

What happens if x or y equals 0?

Q: How would you factor the simplified result -2x^{8}y^{4} ?

Factor out a common monomial, e.g., -2x^{3}y(x^{5}y^{3}) or -2x^{8}y^{4}=(-2x^{4}y^{2})(x^{4}y^{2}), depending on the desired factorization.

Q: How do you divide similar monomials, e.g., \frac{x^{8}y^{4}}{x^{3}y} ?

Subtract exponents: x^{8-3}y^{4-1}=x^{5}y^{3}..

Answer

Multiply coefficients and add exponents: \(x^5y^3(-2x^3y) = -2x^{5+3}y^{3+1} = -2x^8y^4\)

Detailed Explanation

Step-by-step simplification of the expression \(x^{5}y^{3}\bigl(-2x^{3}y\bigr)\):

Rewrite the product by separating numeric and like-base factors:
\[x^{5}y^{3}\bigl(-2x^{3}y\bigr)=(-2)\cdot x^{5}\cdot x^{3}\cdot y^{3}\cdot y^{1}.\]
Here the factor \(y\) is written as \(y^{1}\) to make exponent rules explicit.
Use the rule for multiplying powers with the same base: when multiplying like bases, add the exponents.
For the x-factors:
\[x^{5}\cdot x^{3}=x^{5+3}=x^{8}.\]
For the y-factors:
\[y^{3}\cdot y^{1}=y^{3+1}=y^{4}.\]
Combine the numerical coefficient with the simplified variable parts:
\[(-2)\cdot x^{8}\cdot y^{4}.\]
Write the final simplified result:
\[-2x^{8}y^{4}.\]

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Algebra FAQs

What is the fully simplified form of \(x^{5}y^{3}(-2x^{3}y)\)? Do not use arrows.

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

How do you multiply powers with the same base?

Use \(x^{a}\cdot x^{b}=x^{a+b}\). Here \(x^{5}\cdot x^{3}=x^{8}\) and \(y^{3}\cdot y^{1}=y^{4}\).

How is the negative sign handled in multiplication?

Multiply the coefficients including sign: \(1\cdot(-2)=-2\). The negative stays in front of the final product: \(-2x^{8}y^{4}\)..

What if exponents are zero, e.g., \(x^{0}y^{3}(-2x^{3}y)\)?.

\(x^{0}=1\), so the product becomes \(-2\cdot x^{3}\cdot y^{4} = -2x^{3}y^{4}\)..

Are the exponent rules valid for non-integer exponents?

Yes: \(x^{a}\cdot x^{b}=x^{a+b}\) holds for real \(a,b\) where each expression is defined (watch bases and fractional exponents requiring positive bases for even denominators).

What happens if \(x\) or \(y\) equals \(0\)?

How would you factor the simplified result \( -2x^{8}y^{4} \)?

Factor out a common monomial, e.g., \(-2x^{3}y(x^{5}y^{3})\) or \(-2x^{8}y^{4}=(-2x^{4}y^{2})(x^{4}y^{2})\), depending on the desired factorization.

How do you divide similar monomials, e.g., \( \frac{x^{8}y^{4}}{x^{3}y} \)?

Subtract exponents: \(x^{8-3}y^{4-1}=x^{5}y^{3}\)..

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