Answer:
[tex]\textsf{(a)} \quad 3^{-2}[/tex]
[tex]\textsf{(b)} \quad 3^{-5}[/tex]
[tex]\textsf{(c)} \quad 3^{9}[/tex]
Step-by-step explanation:
Part (a)
Given:
[tex]\dfrac{1}{9}[/tex]
Rewrite 9 as 3²:
[tex]\implies \dfrac{1}{3^2}[/tex]
[tex]\textsf{Apply exponent rule} \quad \dfrac{1}{a^n}=a^{-n}:[/tex]
[tex]\implies 3^{-2}[/tex]
Part (b)
Given:
[tex]\dfrac{3^{-4} \cdot 3^2}{3^3}[/tex]
[tex]\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^{b+c}:[/tex]
[tex]\implies \dfrac{3^{(-4+2)}}{3^3}[/tex]
[tex]\implies \dfrac{3^{-2}}{3^3}[/tex]
[tex]\textsf{Apply exponent rule} \quad \dfrac{a^b}{a^c}=a^{b-c}:[/tex]
[tex]\implies 3^{(-2-3)}[/tex]
[tex]\implies 3^{-5}[/tex]
Part (c)
Given:
[tex]27^2 \div 3^{-3}[/tex]
Rewrite 27 as 3³:
[tex]\implies (3^3)^2 \div 3^{-3}[/tex]
[tex]\textsf{Apply exponent rule} \quad (a^b)^c=a^{bc}:[/tex]
[tex]\implies 3^6 \div 3^{-3}[/tex]
[tex]\textsf{Apply exponent rule} \quad \dfrac{a^b}{a^c}=a^{b-c}[/tex]
[tex]\implies 3^{(6-(-3))}[/tex]
[tex]\implies 3^{(6+3)}[/tex]
[tex]\implies 3^9[/tex]
