Question

A) 25^x = 5^3б) 5^x + 3*5^x-2 = 3500​

Answer 1

Ответ:

Пошаговое объяснение:

Answer 2

Let's solve each equation step by step:

A) $25^x = 5^3$

To solve this equation, we can rewrite both sides of the equation with the same base:

$25 = 5^2$

So, the equation becomes:

$(5^2)^x = 5^3$

Now, we can use the property of exponents that says when you have a power raised to another power, you multiply the exponents:

$5^{2x} = 5^3$

Since the bases (5) are the same, the exponents must be equal:

$2x = 3$

Now, we can solve for $x$:

$x = \frac{3}{2}$

B) $5^x + 3 \cdot 5^{x-2} = 3500$

First, let's simplify the equation by expressing both terms on the left side with the same base:

$5^x + 3 \cdot 5^{x-2} = 3500$

We can rewrite $5^{x-2}$ as $5^x \cdot 5^{-2}$ because of the rule $a^{m-n} = \frac{a^m}{a^n}$:

$5^x + 3 \cdot (5^x \cdot 5^{-2}) = 3500$

Now, distribute the 3 on the left side:

$5^x + 3 \cdot 5^x \cdot 5^{-2} = 3500$

Now, let's simplify further:

$5^x + 3 \cdot \frac{5^x}{5^2} = 3500$

Now, we have a common base (5), and we can combine like terms:

$5^x + \frac{3}{25} \cdot 5^x = 3500$

Now, factor out $5^x$:

$5^x \left(1 + \frac{3}{25}\right) = 3500$

Next, simplify the expression inside the parentheses:

$5^x \left(\frac{28}{25}\right) = 3500$

Now, divide both sides by $\frac{28}{25}$ to solve for $x$:

$5^x = \frac{3500}{\frac{28}{25}}$

Now, simplify the right side:

$5^x = \frac{3500 \cdot 25}{28}$

$5^x = \frac{87500}{28}$

Now, we can calculate the right side:

$5^x = 3125$

Now, we have an equation similar to the first one, so we can use the same method:

$5^x = 5^5$

Since the bases are the same, the exponents must be equal:

$x = 5$

So, the solutions to the equations are:

A) $x = \frac{3}{2}$

B) $x = 5$