Question

if 3^(x+y) = 81 and 81^(x-y)= 3⁸ , then the find values of X and y respectively.


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Answer 1

Answer:

$\huge\boxed{x=3,\ y=1}$

Step-by-step explanation:

$81=3^4\\\\3^{x+y}=81\\\\3^{x+y}=3^4\Rightarrow x+y=4\ (1)\\\\81^{x-y}=3^8\\\\(3^4)^{x-y}=3^8\qquad|\text{use}\ (a^n)^m=a^{nm}\\\\3^{4(x-y)}=3^8\Rightarrow4(x-y)=8\ (2)$

Now we have the system of equations:

$\left\{\begin{array}{ccc}x+y=4\\4(x-y)=8&|\text{divide both sides by 4}\end{array}\right\\\\\underline{+\left\{\begin{array}{ccc}x+y=4\\x-y=2\end{array}\right}\qquad|\text{add both sides of the equations}\\.\qquad 2x=6\qquad|\text{divide both sides by 2}\\.\qquad\boxed{x=3}$

Put it to the first equation:

$3+y=4\qquad|\text{subtract 3 from both sides}\\\\\boxed{y=1}$

Answer 2

Answer:

X = 3 and Y = 1

Step-by-step explanation:

Given,

==> 3 ^ (x + y) = 81 [ Since 81 = 3x3x3x3 = 3⁴ ]

So, we can write 81 as 3⁴.

==> 3 ^ (x + y) = 3⁴ [ if a^m = a^n then m = n ]

If bases are equal then powers are equal.

==> x + y = 4 ---------> 1️⃣ Equation

Also given,

==> 81 ^ (x - y) = 3⁸ [ a^(mn) = (a^m)^n ]

So, 3⁸ = (3⁴)².

==> 81 ^ (x - y) = (3⁴)² [ Since 3⁴ = 81 ]

==> 81 ^ (x - y) = 81² [ if a^m = a^n then m = n ]

If bases are equal then powers are equal.

==> x - y = 2 ---------> 2️⃣ Equation

From 1️⃣ & 2️⃣ equations,

x + y = 4

x - y = 2

--------------

2x = 6

--------------

==> x = 3

1️⃣ ==> x + y = 4

==> 3 + y = 4

==> y = 4 - 3 = 1

==> y = 1

Therefore, X = 3 and Y = 4.

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