Question

1/21 and 1/22
name kayleigh hadathlie
simplify. your answer should contain only positive exponents.

1. $(m^{3})^{-4}$
2. $(p^{10})^{-8}$
3. $(b^{-2})^{7}$
4. $(xy^{-10})^{6}$
5. $(x^{-1}y^{9})^{10}$
6. $(a^{-7})^{-4}$
7. $(x^{-5}y^{9})^{-3}$
8. $(2x^{9}y^{-8})^{8}$
9. $(2v^{-9})^{10}$

Explanation:

Step1: Apply power of a power rule

For $(m^3)^{-4}$: $(a^m)^n=a^{m \cdot n}$, so $m^{3 \cdot (-4)}=m^{-12}$. Convert to positive exponent: $a^{-n}=\frac{1}{a^n}$, so $\frac{1}{m^{12}}$.

Step2: Apply power of a power rule

For $(p^{10})^{-8}$: $p^{10 \cdot (-8)}=p^{-80}$. Convert to positive exponent: $\frac{1}{p^{80}}$.

Step3: Apply power of a power rule

For $(b^{-2})^7$: $b^{-2 \cdot 7}=b^{-14}$. Convert to positive exponent: $\frac{1}{b^{14}}$.

Step4: Apply product & power rules

For $(xy^{-10})^6$: $(ab)^n=a^n b^n$, so $x^6 \cdot (y^{-10})^6$. Then $(y^{-10})^6=y^{-60}$. Convert to positive exponent: $\frac{x^6}{y^{60}}$.

Step5: Apply product & power rules

For $(x^{-1}y^9)^{10}$: $(x^{-1})^{10} \cdot (y^9)^{10}=x^{-10}y^{90}$. Convert to positive exponent: $\frac{y^{90}}{x^{10}}$.

Step6: Apply power of a power rule

For $(a^{-7})^{-4}$: $a^{-7 \cdot (-4)}=a^{28}$.

Step7: Apply product & power rules

For $(x^{-5}y^9)^{-3}$: $(x^{-5})^{-3} \cdot (y^9)^{-3}=x^{15}y^{-27}$. Convert to positive exponent: $\frac{x^{15}}{y^{27}}$.

Step8: Apply product & power rules

For $(2x^9y^{-8})^8$: $2^8 \cdot (x^9)^8 \cdot (y^{-8})^8=256x^{72}y^{-64}$. Convert to positive exponent: $\frac{256x^{72}}{y^{64}}$.

Step9: Apply product & power rules

For $(2v^{-9})^{10}$: $2^{10} \cdot (v^{-9})^{10}=1024v^{-90}$. Convert to positive exponent: $\frac{1024}{v^{90}}$.

Answer:

1. $\frac{1}{m^{12}}$
2. $\frac{1}{p^{80}}$
3. $\frac{1}{b^{14}}$
4. $\frac{x^6}{y^{60}}$
5. $\frac{y^{90}}{x^{10}}$
6. $a^{28}$
7. $\frac{x^{15}}{y^{27}}$
8. $\frac{256x^{72}}{y^{64}}$
9. $\frac{1024}{v^{90}}$