Question

(1/6)^x+(1/8)^x=(0,1)^x

Answer 1

(1/6)ˣ + (1/8)ˣ = (1/10)ˣ

Разделим обе части уравнения на (1/10)ˣ

(1/6)ˣ : (1/10)ˣ + (1/8)ˣ : (1/10)ˣ = (1/10)ˣ : (1/10)ˣ

(5/3)ˣ + (5/4)ˣ = 1

В левой части данного уравнения функция монотонно возрастает, а в правой части - константа. Значит, уравнение имеет не более одного корня. Попробуем подобрать корень, перебирая небольшие по модулю целые числа. Число х = - 2 является корнем уравнения. Других корней уравнение не имеет.

ОТВЕТ: - 2

Answer 2

To solve the equation (1/6)^x + (1/8)^x = (0.1)^x, we can simplify the fractions and rewrite them with common bases.

Let's rewrite (1/6)^x as (6^(-1))^x, (1/8)^x as (8^(-1))^x, and (0.1)^x as (10^(-1))^x.

Now the equation becomes: (6^(-1))^x + (8^(-1))^x = (10^(-1))^x

Using the property (a^b)^c = a^(b*c), we can simplify further: 6^(-x) + 8^(-x) = 10^(-x)

To proceed, let's find a common denominator for the terms on the left side. The common denominator is 6^x * 8^x, so we can rewrite the equation as: (8^x + 6^x)/(6^x * 8^x) = 10^(-x)

Now, multiplying both sides of the equation by (6^x * 8^x), we get: 8^x + 6^x = (10^(-x)) * (6^x * 8^x)

Since (10^(-x)) * (6^x * 8^x) = ((10 * 6 * 8)^(-x)) = (480^(-x)), we can rewrite the equation as: 8^x + 6^x = 480^(-x)

Unfortunately, this equation does not have a simple algebraic solution. It may be possible to approximate the value of x using numerical methods or graphing calculators.