Question

\\(\frac{(-5s^4)^2}{15s^4}\\) \\(\frac{12k^{-10}}{(3k)^4}\\)

\\(\frac{(4x^{-5}k^3)^{-5}}{12x^{-6}k^{-8}}\\) \\(\frac{6x^2t}{(-2x^{-3}t^4)^5}\\)

\\(\frac{-3(u^{-4})^2}{6a^{-9}}\\) \\(\frac{15u^{10}q^{-6}}{(-3u^4q^3)^3}\\)

Explanation:

Step1: Simplify \(\frac{(-5s^{4})^{2}}{15s^{4}}\)

First, apply the power of a product rule \((ab)^n = a^n b^n\) to the numerator: \((-5s^{4})^{2}=(-5)^{2}(s^{4})^{2}=25s^{8}\). Then, divide by the denominator: \(\frac{25s^{8}}{15s^{4}}\). Simplify the coefficients and use the quotient rule for exponents \( \frac{a^m}{a^n}=a^{m - n}\): \(\frac{25}{15}s^{8 - 4}=\frac{5}{3}s^{4}\).

Step2: Simplify \(\frac{12k^{-10}}{(3k)^{4}}\)

First, expand the denominator using \((ab)^n=a^n b^n\): \((3k)^{4}=3^{4}k^{4}=81k^{4}\). Then, rewrite \(k^{-10}\) as \(\frac{1}{k^{10}}\) (though we can use the quotient rule directly). So, \(\frac{12k^{-10}}{81k^{4}}=\frac{12}{81}k^{-10 - 4}\). Simplify the coefficient: \(\frac{4}{27}k^{-14}=\frac{4}{27k^{14}}\).

Step3: Simplify \(\frac{(4x^{-5}k^{3})^{-5}}{12x^{-6}k^{-8}}\)

First, apply the power of a product rule to the numerator: \((4x^{-5}k^{3})^{-5}=4^{-5}(x^{-5})^{-5}(k^{3})^{-5}=4^{-5}x^{25}k^{-15}\). Then, divide by the denominator: \(\frac{4^{-5}x^{25}k^{-15}}{12x^{-6}k^{-8}}\). Use the quotient rule for exponents: \(4^{-5}x^{25-(-6)}k^{-15 - (-8)}\div12\). Simplify exponents: \(4^{-5}x^{31}k^{-7}\div12\). Rewrite \(4^{-5}=\frac{1}{4^{5}}=\frac{1}{1024}\), so \(\frac{1}{1024\times12}x^{31}k^{-7}=\frac{1}{12288}x^{31}\frac{1}{k^{7}}=\frac{x^{31}}{12288k^{7}}\).

Step4: Simplify \(\frac{6x^{2}t}{(-2x^{-3}t^{4})^{5}}\)

First, expand the denominator: \((-2x^{-3}t^{4})^{5}=(-2)^{5}(x^{-3})^{5}(t^{4})^{5}=-32x^{-15}t^{20}\). Then, divide: \(\frac{6x^{2}t}{-32x^{-15}t^{20}}\). Simplify coefficients and use quotient rule: \(\frac{6}{-32}x^{2-(-15)}t^{1 - 20}=-\frac{3}{16}x^{17}t^{-19}=-\frac{3x^{17}}{16t^{19}}\).

Step5: Simplify \(\frac{-3(u^{-4})^{2}}{6a^{-9}}\)

First, simplify the numerator: \((u^{-4})^{2}=u^{-8}\), so numerator is \(-3u^{-8}\). Then, divide by denominator: \(\frac{-3u^{-8}}{6a^{-9}}\). Simplify coefficients and use negative exponents: \(-\frac{1}{2}u^{-8}a^{9}=-\frac{a^{9}}{2u^{8}}\).

Step6: Simplify \(\frac{15u^{10}q^{-6}}{(-3u^{4}q^{3})^{3}}\)

First, expand the denominator: \((-3u^{4}q^{3})^{3}=(-3)^{3}(u^{4})^{3}(q^{3})^{3}=-27u^{12}q^{9}\). Then, divide: \(\frac{15u^{10}q^{-6}}{-27u^{12}q^{9}}\). Simplify coefficients and use quotient rule: \(\frac{15}{-27}u^{10 - 12}q^{-6 - 9}=-\frac{5}{9}u^{-2}q^{-15}=-\frac{5}{9u^{2}q^{15}}\).

Answer:

1. \(\frac{5}{3}s^{4}\)
2. \(\frac{4}{27k^{14}}\)
3. \(\frac{x^{31}}{12288k^{7}}\)
4. \(-\frac{3x^{17}}{16t^{19}}\)
5. \(-\frac{a^{9}}{2u^{8}}\)
6. \(-\frac{5}{9u^{2}q^{15}}\)