Q. Fully simplify. \[ (2x^3 y^5)^5 = 2^5 x^{15} y^{25} = 32 x^{15} y^{25} \]

Q: How do you apply an exponent to a product like (2x^3y^5)^5 ?.

Use (ab)^n = a^n b^n. So (2x^3y^5)^5 = 2^5 (x^3)^5 (y^5)^5..

Q: How do you handle a power raised to a power?

Use (x^a)^b = x^{ab}. Example: (x^3)^5 = x^{15}.

Q: What if the outer exponent were 0, e.g. (2x^3y^5)^0?

What if the outer exponent were 0, e.g. (2x^3y^5)^0?

Q: What if the base has a negative sign, e.g. (-2x^3y^5)^5 ?.

Include the sign: (-2)^5 = -32, so (-2x^3y^5)^5 = -32x^{15}y^{25}..

Q: How do fractional outer exponents work, like (x^3)^{1/5} ?

Use (x^a)^{b} = x^{ab}, so (x^3)^{1/5} = x^{3/5}. For even roots, check domain (real vs complex).

Answer

Apply the power to each factor: \( (2x^3y^5)^5 = 2^5(x^3)^5(y^5)^5 = 32x^{15}y^{25} \)

Detailed Explanation

Problem

Simplify the expression

\( (2x^{3}y^{5})^{5} \)

Step-by-step explanation

Apply the power of a product rule: when a product is raised to an exponent, raise each factor to that exponent. Also use the power-of-a-power rule for powers of variables. Thus
\( (2x^{3}y^{5})^{5} = 2^{5}\,(x^{3})^{5}\,(y^{5})^{5} \)
Simplify each factor separately.
- Compute the numerical power: \(2^{5} = 32\).
- Use the power-of-a-power rule: \((x^{3})^{5} = x^{3\cdot 5} = x^{15}\).
- Similarly \((y^{5})^{5} = y^{5\cdot 5} = y^{25}\).
Combine the simplified factors (they are all multiplied):
\( 2^{5}\,(x^{3})^{5}\,(y^{5})^{5} = 32\,x^{15}\,y^{25} \)
There are no like bases left to combine, so the expression is fully simplified.

Final answer

\[
32\,x^{15}\,y^{25}
\]

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Algebra FAQs

How do you apply an exponent to a product like \( (2x^3y^5)^5 \)?.

Use \( (ab)^n = a^n b^n\). So \( (2x^3y^5)^5 = 2^5 (x^3)^5 (y^5)^5\)..

How do you handle a power raised to a power?

Use \( (x^a)^b = x^{ab}\). Example: \( (x^3)^5 = x^{15}\).

What is \(2^5\)?.

\(2^5 = 32\).

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

\(32x^{15}y^{25}\)..

What common mistakes should I avoid?

Don’t add exponents; multiply them. Don’t forget to raise the numerical coefficient. Keep parentheses so the outer exponent applies to every factor.

What if the outer exponent were 0, e.g. \( (2x^3y^5)^0\)?

What if the base has a negative sign, e.g. \( (-2x^3y^5)^5 \)?.

Include the sign: \( (-2)^5 = -32\), so \( (-2x^3y^5)^5 = -32x^{15}y^{25}\)..

How do fractional outer exponents work, like \( (x^3)^{1/5} \)?

Use \( (x^a)^{b} = x^{ab}\), so \( (x^3)^{1/5} = x^{3/5}\). For even roots, check domain (real vs complex).

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