Question

1. $(2b^4)^{-1}$
2. $(x^2y^{-1})^2$
3. $(2x^4y^{-3})^{-1}$
4. $(3m)^{-2}$
5. $\frac{r^2}{2r^3}$
6. $\frac{x^{-1}}{4x^4}$
7. $\frac{3n^4}{3n^3}$
8. $\frac{m^4}{2m^4}$
9. $\frac{3m^{-4}}{m^3}$
10. $\frac{2x^4y^{-4}z^{-3}}{3x^2y^{-3}z^4}$
11. $\frac{4x^0y^{-2}z^3}{4x}$
12. $\frac{2h^3j^{-3}k^4}{3jk}$
13. $\frac{4m^4n^3p^3}{3m^2n^2p^4}$
14. $\frac{3x^3y^{-1}z^{-1}}{x^{-4}y^0z^0}$

Explanation:

Let's solve these problems one by one using the properties of exponents.

Problem 17: \((2b^{4})^{-1}\)

Step 1: Apply the power of a product rule \((ab)^n = a^n b^n\)

\((2b^{4})^{-1} = 2^{-1}(b^{4})^{-1}\)

Step 2: Apply the negative exponent rule \(a^{-n} = \frac{1}{a^n}\) and the power of a power rule \((a^m)^n = a^{mn}\)

\(2^{-1} = \frac{1}{2}\) and \((b^{4})^{-1} = b^{-4} = \frac{1}{b^4}\)
So, \((2b^{4})^{-1} = \frac{1}{2b^4}\)

Step 1: Apply the power of a product rule \((ab)^n = a^n b^n\)

\((x^{2}y^{-1})^{2} = (x^{2})^{2}(y^{-1})^{2}\)

Step 2: Apply the power of a power rule \((a^m)^n = a^{mn}\)

\((x^{2})^{2} = x^{4}\) and \((y^{-1})^{2} = y^{-2} = \frac{1}{y^2}\)
So, \((x^{2}y^{-1})^{2} = \frac{x^4}{y^2}\)

Step 1: Apply the power of a product rule \((ab)^n = a^n b^n\)

\((2x^{4}y^{-3})^{-1} = 2^{-1}(x^{4})^{-1}(y^{-3})^{-1}\)

Step 2: Apply the negative exponent rule and the power of a power rule

\(2^{-1} = \frac{1}{2}\), \((x^{4})^{-1} = x^{-4} = \frac{1}{x^4}\), \((y^{-3})^{-1} = y^{3}\)
So, \((2x^{4}y^{-3})^{-1} = \frac{y^3}{2x^4}\)

Answer:

\(\frac{1}{2b^4}\)

Problem 18: \((x^{2}y^{-1})^{2}\)