Question

1. simplify

$\frac{x^{-3}}{x^{5}}$

1. simplify

$\frac{m^{-6}}{m^{-4}}$

1. simplify

$\frac{20x^{-10}y^{12}z^{-2}}{6x^{-15}y^{8}z^{4}}$

1. simplify

$\frac{32a^{-6}b^{4}c^{10}}{18a^{-18}b^{6}c^{-3}}$

1. simplify

$(144k^{2}r^{12})^{\frac{1}{2}}$

1. simplify

$(64x^{4}y^{16})^{\frac{1}{2}}$

Explanation:

Step1: Subtract exponents (same base)

$\frac{x^{-3}}{x^5}=x^{-3-5}=x^{-8}=\frac{1}{x^8}$

Step2: Subtract exponents (same base)

$\frac{m^{-6}}{m^{-4}}=m^{-6-(-4)}=m^{-2}=\frac{1}{m^2}$

Step3: Simplify coefficients + exponents

$\frac{20x^{-10}y^{12}z^{-2}}{6x^{-15}y^8z^4}=\frac{10}{3}x^{-10-(-15)}y^{12-8}z^{-2-4}=\frac{10}{3}x^{5}y^{4}z^{-6}=\frac{10x^5y^4}{3z^6}$

Step4: Simplify coefficients + exponents

$\frac{32a^{-6}b^4c^{10}}{18a^{-18}b^6c^{-3}}=\frac{16}{9}a^{-6-(-18)}b^{4-6}c^{10-(-3)}=\frac{16}{9}a^{12}b^{-2}c^{13}=\frac{16a^{12}c^{13}}{9b^2}$

Step5: Apply square root to each term

$(144k^2r^{12})^\frac{1}{2}=\sqrt{144}\cdot\sqrt{k^2}\cdot\sqrt{r^{12}}=12kr^6$

Step6: Apply square root to each term

$(64x^4y^{16})^\frac{1}{2}=\sqrt{64}\cdot\sqrt{x^4}\cdot\sqrt{y^{16}}=8x^2y^8$

Answer:

1. $\frac{1}{x^8}$
2. $\frac{1}{m^2}$
3. $\frac{10x^5y^4}{3z^6}$
4. $\frac{16a^{12}c^{13}}{9b^2}$
5. $12kr^6$
6. $8x^2y^8$