Question

simplify the expression.
$(4m^2n^5)^3$
fill in the blank 12.5 points
simplify the expression.
$(-3x^7y^3z^0)^5$

Explanation:

Step1: Apply power rule to first term

Use $(ab)^n=a^n b^n$ and $(a^m)^n=a^{mn}$:
$$(4m^2n^5)^3 = 4^3 \cdot (m^2)^3 \cdot (n^5)^3$$

Step2: Calculate each part of first term

Compute exponents and constants:
$$4^3=64,\ (m^2)^3=m^{6},\ (n^5)^3=n^{15}$$

Step3: Simplify second term's zero exponent

Use $z^0=1$:
$$(-3x^7y^3z^0)^5 = (-3x^7y^3 \cdot 1)^5 = (-3x^7y^3)^5$$

Step4: Apply power rule to second term

Use $(ab)^n=a^n b^n$ and $(a^m)^n=a^{mn}$:
$$(-3x^7y^3)^5 = (-3)^5 \cdot (x^7)^5 \cdot (y^3)^5$$

Step5: Calculate each part of second term

Compute exponents and constants:
$$(-3)^5=-243,\ (x^7)^5=x^{35},\ (y^3)^5=y^{15}$$

Answer:

For $(4m^2n^5)^3$: 64, 6, 15
For $(-3x^7y^3z^0)^5$: -243, 35, 15