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

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

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

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

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

Q: How would it change if the second factor were 3x^3 instead of x^3?

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

Q: What if one exponent were zero, e.g., x^0?

Since x^0=1, multiplying by x^0 leaves the other factor unchanged.

Q: How do negative or fractional exponents behave when multiplying?

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

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

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 \(x^3\)?

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 \(-2x^3\)?

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 \(x^3\)?

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

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 \(x^3\)?

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

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

How do negative or fractional exponents behave when multiplying?

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

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