Answer:
The value of a is -4
Step-by-step explanation:
[tex](\frac{1}{9}) ^{a+1}[/tex] = [tex]81^{a+1}[/tex] × [tex]27^{2-a}[/tex] ------------------------------------(1)
No solve this, we need to take note that 9 = 3² , 81 = [tex]3^{4}[/tex] and 27 = [tex]3^{3}[/tex]
Now, we are going to replace 9 , 81 and 27 by 3², [tex]3^{4}[/tex] and [tex]3^{3}[/tex] respectively in equation(1)
[tex](\frac{1}{3^{2} }) ^{a+1}[/tex] = [tex](3^{4}) ^{a+1}[/tex] × [tex](3^{3} )^{2-a}[/tex] ------------------------------(2)
also [tex]\frac{1}{3^{2} }[/tex] = [tex]3^{-2}[/tex]
we are going to replace [tex]\frac{1}{3^{2} }[/tex] by [tex]3^{-2}[/tex] in equation (2)
[tex](3^{-2})^{a+1}[/tex] = [tex](3^{4}) ^{a+1}[/tex] × [tex](3^{3} )^{2-a}[/tex]
We can now open the parenthesis
[tex]3^{-2a-2}[/tex] = [tex]3^{4a + 4}[/tex] × [tex]3^{6-3a}[/tex]
At the right-hand side of the equation, we will apply the law of indices that state [tex]x^{a}[/tex] × [tex]x^{b}[/tex] = [tex]x^{a+b}[/tex]
This implies we will take just 3 and then add-up all the powers
[tex]3^{-2a-2}[/tex] = [tex]3^{4a + 4+ 6-3a}[/tex]
The 3 on the left-hand side will cancel-out the 3 on the right-hand side leaving us with just the powers
-2a - 2 = 4a + 4 + 6 - 3a
-2a - 2 = 4a + 10 - 3a
collect like-term
Which means we will take all the digits with variable to the left-hand side and then take all the digits standing alone to the right-hand side of the equation
-2a - 4a + 3a = 10 + 2
-6a + 3a = 12
-3a = 12
Divide both-side of the equation by -3
[tex]\frac{-3a}{-3}[/tex] = [tex]\frac{12}{-3}[/tex]
a = -4
Therefore, the value of a is -4
