Question

Если 9^x+9^-x=14 найдите 3^x+3^-x

Answer 1

(3ˣ+3⁻ˣ)²=3²ˣ+2*3ˣ*3⁻ˣ+3⁻²ˣ=9ˣ+2+9⁻ˣ=(9ˣ+9⁻ˣ)+2=14+2=16     ⇒

(3ˣ+3⁻ˣ)²=16

3ˣ+3⁻ˣ=√16=+/-4.

√16 будет положительный, так как  3ˣ>0 и 3⁻ˣ>0.

Ответ: 3ˣ+3⁻ˣ=4.

Answer 2

To find the value of 3^x + 3^(-x), we can make use of a substitution to simplify the equation. Let's set y = 3^x + 3^(-x).

Now, notice that we have the following relationship:

(1) 9^x + 9^(-x) = 14

We can rewrite 9 as (3^2) since 9 is equal to 3 raised to the power of 2:

(2) (3^2)^x + (3^2)^(-x) = 14

Using the properties of exponents, (a^m)^n is equal to a^(m*n), we can simplify (2):

(3) 3^(2x) + 3^(-2x) = 14

Now, let's express 3^(-2x) in terms of y. Notice that 3^(-2x) is the reciprocal of y:

(4) 3^(-2x) = 1/y

Substitute equation (4) back into equation (3):

3^(2x) + 1/y = 14

Now, let's express 3^(2x) in terms of y. Notice that 3^(2x) is equal to y^2:

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

Substitute equation (5) back into the modified equation:

y^2 + 1/y = 14

Now, we have a quadratic equation in terms of y. To solve for y, we can multiply the entire equation by y to eliminate the fraction:

y^3 + 1 = 14y

Rearrange the equation:

y^3 - 14y + 1 = 0

This is a cubic equation in terms of y, and solving it for an exact value of y might be challenging. However, we can still use numerical methods or calculators to find an approximate value of y. Once we have an approximate value for y, we can find 3^x + 3^(-x) by substituting back the value of y.

Please note that the exact value of y may not have a simple expression, and numerical methods might be necessary to approximate the result.