Answer:
Step-by-step explanation:
We will use division, the order of operations, and the rules of exponents to simplify the given expression. Here are all of the steps to simplify.
Given:
[tex]\displaystyle (\frac{2a^3b^2c^7}{3a^2b^3c^7})^{-2}[/tex]
Something divided by itself is equal to 1:
[tex]\displaystyle (\frac{2a^3b^2*c^7}{3a^2b^3*c^7})^{-2}[/tex]
[tex]\displaystyle (\frac{2a^3b^2}{3a^2b^3})^{-2}[/tex]
Exponents quotient rule:
➜ If the bases are the same, subtract the exponents: [tex]\displaystyle \frac{a^m}{a^n} =a^{m-n}[/tex]
[tex]\displaystyle (\frac{2*a^3*b^2}{3*a^2*b^3})^{-2}[/tex]
[tex]\displaystyle (\frac{2*a^{3-2}*b^{2-3}}{3})^{-2}[/tex]
[tex]\displaystyle (\frac{2ab^{-1}}{3})^{-2}[/tex]
Negative exponent rule:
➜ [tex]\displaystyle a^{-m}=\frac{1}{a^m}[/tex]
[tex]\displaystyle (\frac{2a}{3b})^{-2}[/tex]
[tex]\displaystyle \frac{1}{(\dfrac{2a}{3b})^{2}}[/tex]
Square:
[tex]\displaystyle \frac{1}{(\dfrac{2a}{3b})(\dfrac{2a}{3b})}[/tex]
[tex]\displaystyle \frac{1}{(\dfrac{4a^2}{9b^2})}[/tex]
Division of fractions:
➜ Remember "keep, change, flip"
[tex]\displaystyle \frac{\dfrac{1}{1} }{(\dfrac{4a^2}{9b^2})}[/tex]
[tex]\displaystyle \frac{1}{1} *\dfrac{9b^2}{4a^2}[/tex]
[tex]\displaystyle \dfrac{9b^2}{4a^2}[/tex]
