Question

3^(x+3)=0.75*2^(3x+8)

Answer 1

$\displaystyle \tt 3^{x+3}=0,75\cdot2^{3x+8}\\\\27\cdot3^{x}=0,75\cdot2^{8}\cdot2^{3x}\\\\27\cdot3^{x}=192\cdot2^{3x}\\\\\frac{27\cdot3^{x}}{8^{x}}=192\\\\\\\bigg(\frac{3}{8}\bigg)^{x}=\frac{192}{27}\\\\\\ \bigg(\frac{3}{8}\bigg)^{x}=\frac{64}{9}\\\\\\\bigg(\frac{3}{8}\bigg)^{x}=\bigg(\frac{8}{3}\bigg)^{2}\\\\\\x=-2$

Ответ: {-2}

Answer 2

To solve the given equation $3^{x+3} = 0.75 \times 2^{3x+8}$, we can start by rewriting $0.75$ as $\frac{3}{4}$ to work with powers of $3$ and $2$ only:

$3^{x+3} = \frac{3}{4} \times 2^{3x+8}$.

Now, let's manipulate the equation to isolate the variables:

$3^{x+3} = \frac{3}{4} \times 2^{3x+8}$

We know that $0.75 = \frac{3}{4}$, so we can rewrite the right side:

$3^{x+3} = \frac{3}{4} \times 2^{3x+8}$ $3^{x+3} = \frac{3}{4} \times 2^{3x} \times 2^8$ $3^{x+3} = \frac{3}{4} \times 2^8 \times 2^{3x}$ $3^{x+3} = 2^8 \times \frac{3}{4} \times 2^{3x}$ $3^{x+3} = 2^8 \times \frac{3}{4} \times (2^3)^x$

Now, we can use the properties of exponents to rewrite the equation:

$3^{x+3} = 2^8 \times \frac{3}{4} \times 2^{3x}$ $3^{x+3} = 2^8 \times \frac{3}{4} \times 2^{3x}$ $3^{x+3} = 2^8 \times \frac{3}{4} \times 8^x$

Since $8 = 2^3$, we can further simplify the equation:

$3^{x+3} = 2^8 \times \frac{3}{4} \times (2^3)^x$ $3^{x+3} = 2^8 \times \frac{3}{4} \times 2^{3x}$ $3^{x+3} = 2^{8 + 3x} \times \frac{3}{4}$

Now, we can equate the exponents:

$x+3 = 8 + 3x$

Now, solve for $x$:

$x - 3x = 8 - 3$ $-2x = 5$

Divide by $-2$ to find $x$:

$x = \frac{-5}{2}$

So, the value of $x$ that satisfies the given equation is $x = -\frac{5}{2}$.