Lets start by rewriting everything in terms of 3's
81 = (3⁴ )
27 = (3³)
9 = (3²)
45 = 3² x 5
Now lets substitute these into 81^7-27^9-9^13:
81⁷ - 27⁹ -9¹³
= (3⁴)⁷ - (3³)⁹ - (3²)¹³ ---> (use your indices rules to simplify)
= 3²⁸ - 3²⁷ - 3²⁶
Note: To prove this is divisible by 45, it has to be divisible by both 3 and 5.
Let's start by taking out the largest factor of 3 as possible from 3²⁸ - 3²⁷ - 3²⁶, which is 3²⁶:
3²⁸ - 3²⁷ - 3²⁶
= 3²⁶ ( 3² - 3¹ - 3⁰) --> (Now we have something we can easily simplify)
= 3²⁶ ( 9 - 3 - 1)
= 3²⁶ (5)
Oh look! This is clearly divisible by 5, and also by 3², that means it is divisible by 45!
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So:
81⁷ - 27⁹ -9¹³ ÷ 45
= 3²⁶ (5) ÷ (3² x 5)
= 3²⁶ ÷ 3²
= 3²⁴
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Following are the calculation to the given value:
Given:
[tex]\to 81^7-27^9-9^{13} \div 45\\\\[/tex]
To find:
divsion=?
Solution:
[tex]\to 81^7-27^9-9^{13} \div 45\\[/tex]
Converting each expression into 3 to the power then:
[tex]\to 81^7=3^{28}\\\\\to 27^9=3^{27}\\\\\to 9^{13}=3^{26}\\\\[/tex]
Taking the common [tex]3^{26}[/tex]from the above expression:
[tex]\to 3^{26} \times (3^2- 3- 1)\\\\\to 3^{26} \times(9-3-1)\\\\\to 3^{26} \times 5[/tex]
Therefore the given question resolves: [tex]\frac{(3^{26} \times 5)}{45}[/tex] is divisible or not 45 comprise of
[tex]3^{2} \times 5[/tex]
Since the power of the divisor is less than the dividend so, the equation is divisible that is [tex]3^{24}[/tex].
Learn more:
brainly.com/question/8450189
