Question

1. simplify the expression \\(\frac{(m^5n)^{-2}(m^2n^{-2})^3}{m^{-1}n^{-2}}\\) by using positive exponents.

a. \\(\frac{m^7}{n^6}\\)
b. \\(\frac{1}{m^7n^6}\\)
c. \\(\frac{m^5}{n^{10}}\\)
d. \\(\frac{1}{m^5n^6}\\)

Explanation:

Step1: Expand numerator terms

Apply power rule $(x^a)^b=x^{ab}$:
$$(m^6n)^{-2}=m^{6\times(-2)}n^{1\times(-2)}=m^{-12}n^{-2}$$
$$(m^2n^{-2})^3=m^{2\times3}n^{-2\times3}=m^6n^{-6}$$
Multiply numerator terms: $m^{-12}n^{-2} \times m^6n^{-6}=m^{-12+6}n^{-2+(-6)}=m^{-6}n^{-8}$

Step2: Rewrite division as multiplication

Use $x^{-a}=\frac{1}{x^a}$, so $\frac{1}{m^{-1}n^{-2}}=m^{1}n^{2}$:
$$m^{-6}n^{-8} \times m^{1}n^{2}$$

Step3: Combine like bases

Add exponents for same bases:
$$m^{-6+1}n^{-8+2}=m^{-5}n^{-6}$$

Step4: Convert to positive exponents

Apply $x^{-a}=\frac{1}{x^a}$:
$$m^{-5}n^{-6}=\frac{1}{m^5n^6}$$

Answer:

D. $\frac{1}{m^5n^6}$