Question

4^x+1 + 4*3^x = 3^x+2 - 4^x

Answer 1

4^x*4+4*3^x=3^x*3^2-4^x

4*4^x+4^x=9*3^x-4*3^x

5*4^x=5*3^x

5*(4/3)^x=5

(4/3)^x=1^0

x=0

Answer 2

To solve the equation $4^{x+1} + 4 \cdot 3^x = 3^{x+2} - 4^x$, we can start by isolating the terms with the same base on each side of the equation. In this case, we have 4 and 3 as the bases. Let's simplify the equation step by step:

$4^{x+1} + 4 \cdot 3^x = 3^{x+2} - 4^x$

First, we can subtract $4^x$ from both sides to move all terms with $4$ to one side of the equation:

$4^{x+1} + 4 \cdot 3^x + 4^x = 3^{x+2}$

Now, let's use the properties of exponents to rewrite $4^{x+1}$ as $4 \cdot 4^x$:

$4 \cdot 4^x + 4 \cdot 3^x + 4^x = 3^{x+2}$

Next, we can factor out $4^x$ from the terms on the left side:

$4^x(4 + 1) + 4 \cdot 3^x = 3^{x+2}$

Simplify further:

$4^x \cdot 5 + 4 \cdot 3^x = 3^{x+2}$

Now, we can rewrite $3^{x+2}$ as $3^x \cdot 3^2$ using the properties of exponents:

$4^x \cdot 5 + 4 \cdot 3^x = 9 \cdot 3^x$

We have successfully isolated the terms with $3^x$ on one side of the equation. Now, we can move all other terms to the right side:

$4^x \cdot 5 - 4 \cdot 3^x = 9 \cdot 3^x - 4^x$

Now, the equation is in a form where we can combine like terms:

$5 \cdot 4^x - 4 \cdot 3^x = 9 \cdot 3^x - 4^x$

Now, let's add $4^x$ to both sides to collect the $4^x$ terms on the left side:

$5 \cdot 4^x - 4^x - 4 \cdot 3^x = 9 \cdot 3^x$

$4^x(5 - 1) - 4 \cdot 3^x = 9 \cdot 3^x$

$4^x \cdot 4 - 4 \cdot 3^x = 9 \cdot 3^x$

Now, we can simplify further:

$4^x \cdot 4 - 4 \cdot 3^x = 9 \cdot 3^x$

$4^x \cdot 4 - 4 \cdot 3^x = 3^x \cdot 9$

Now, we have the equation in terms of $4^x$ and $3^x$. Let's simplify it further:

$4^x \cdot 4 - 4 \cdot 3^x = 3^x \cdot 9$

$4^x \cdot 4 - 4 \cdot 3^x = 3^x \cdot (3 \cdot 3)$

$4^x \cdot 4 - 4 \cdot 3^x = 3^x \cdot 3^2$