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

Answer

Assuming the expression is \(-2x^3y^5(x^3)\):

Multiply coefficients and add exponents for like bases:
\[
-2x^3y^5(x^3)=-2\cdot x^{3+3}y^5=-2x^6y^5.
\]

Final result: \(-2x^6y^5\).

Detailed Explanation

Write the given expression clearly:

\[ -2x^{3}y^{5}\,(x^{3}) \]
Use the rule for multiplying powers with the same base: \(a^{m}\cdot a^{n}=a^{m+n}\).
For the \(x\)-terms, \(x^{3}\cdot x^{3}=x^{3+3}=x^{6}\). The coefficient \(-2\) and \(y^{5}\) remain unchanged.
Combine the parts to get the simplified result:

\[ -2x^{6}y^{5} \]

Final answer: \[ -2x^{6}y^{5} \]

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FAQs

What is the simplified form of \(-2x^3y^5(x^3)\)?

-A: Multiply coefficients and add exponents for like bases: \(-2\cdot1=-2\), \(x^{3+3}=x^6\). Result: \(-2x^6y^5\).

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

-A: Because \(a^m\cdot a^n=a^{m+n}\). Multiplying powers with the same base combines their exponents.

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

-A: The negative is part of the coefficient. Multiply it with other coefficients; here \(-2\cdot1=-2\), so the product stays negative.

What happens to \(y^5\) when multiplying by \(x^3\)?

-A: If a variable does not appear in the other factor, its exponent is unchanged. So \(y^5\) remains \(y^5\).

Does multiplication order matter for monomials like \(-2x^3y^5\cdot x^3\)?

-A: No. Multiplication is commutative and associative; you can multiply coefficients and like bases in any order.

How would it change if the second factor were \(3x^3\) instead of \(x^3\)?

-A: Multiply coefficients: \(-2\cdot3=-6\). Combine \(x\) exponents: \(x^{3+3}=x^6\). Result: \(-6x^6y^5\).

What if one exponent were zero, e.g. \(x^0\)?

-A: Since \(x^0=1\), multiplying by \(x^0\) leaves the other factor unchanged.

How do negative or fractional exponents behave when multiplying?

-A: Same rule: add exponents. Example: \(x^{1/2}\cdot x^{-3/2}=x^{-1}\) because \(1/2+(-3/2)=-1\).

How do you divide similar monomials?

-A: Use subtraction: \(\dfrac{x^m}{x^n}=x^{m-n}\) (if \(x\neq0\)). -Also divide coefficients normally.

What are common mistakes to avoid?

-A: Forgetting to add exponents, trying to multiply unlike bases (e.g., \(x\) and \(y\)), or mishandling the sign of the coefficient.

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