Respuesta :

gmany

Answer:

[tex]\large\boxed{268=3^5+5^2}[/tex]

Step-by-step explanation:

Consecutive power of number 3:

[tex]3^1=3\\3^2=9\\3^3=27\\3^4=81\\3^5=243\\3^6=729>268[/tex]

[tex]268-243=25=5^2[/tex]

Therefore

[tex]268=243+25=3^5+5^2[/tex]

Let's check the others:

[tex]268-81=187\leftarrow\text{it's not a square}\\\\268-27=241\leftarrow\text{it's not a square}\\\\268-9=259\leftarrow\text{it's not a square}\\\\268-3=265\leftarrow\text{it's not a square}\\\\[/tex]

Therefore we have only one solution.

Since 268 is not much big number,  we can try manually getting power of 3 and seeing if the rest of the quantity is a square.

268 can be written as  [tex]268 = 3^5 + 5^2[/tex]

How to know what power of 3 will be used?

We can try manually getting power of 3 and seeing if the rest of the quantity is a square. We can proceed as follows.

[tex]268 = 3^a + x\\x = 268 - 3^a[/tex]

x needs to be perfect square(a positive number), thus we will proceed to put values in 'a' such that x stays being positive.

Putting values of a

[tex]x = 268 - 3^1 = 268 - 3 = 265\\x = 268 - 3^2 = 268 - 9 = 259\\x = 268 - 3^3 = 268 - 27 = 241\\x = 268 - 3^4 = 268 - 81 = 187\\x = 268 - 3^5 = 268 - 243 = 25\\x = 268 - 3^6 = 268 - 486 < 0[/tex]

Thus, only for a = 5, it works since from the obtained value of x, only 25 is perfect square (square of integer quantity).

Thus, we have:

[tex]268 = 3^5 + 5^2[/tex]

Learn more about perfect squares here:

https://brainly.com/question/997406

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