Question

8. Функцію задано формулою у=х^а 1) Обчисліть показник степеня, якщо а = 4^-5/8 *2^1,5 2) Побудуйте ескiз графіка функції у=х^а Наведіть повне розв'язання задач 9 і 10.

Answer 1

Calculation of the Exponent:

To calculate the exponent (a) in the equation u = x^a, we are given the value of a as 4^(-5/8) * 2^(1.5).

To simplify this expression, we can start by evaluating the exponents separately and then multiplying the results.

Let's calculate each exponent step by step:

1. Evaluating the exponent 4^(-5/8): - The exponent 4^(-5/8) means taking the reciprocal of the 8th root of 4 raised to the power of 5. - We can rewrite 4^(-5/8) as (4^(1/8))^(-5). - Taking the 8th root of 4 gives us 4^(1/8) ≈ 1.189207. - Now, raising 1.189207 to the power of -5 gives us approximately 0.672413.

2. Evaluating the exponent 2^(1.5): - The exponent 2^(1.5) means taking the square root of 2 raised to the power of 3/2. - Taking the square root of 2 gives us 2^(1/2) ≈ 1.414214. - Now, raising 1.414214 to the power of 3 gives us approximately 2.828427.

3. Multiplying the results: - Multiplying 0.672413 and 2.828427 gives us approximately 1.901307.

Therefore, the value of the exponent (a) is approximately 1.901307.

Sketching the Graph:

To sketch the graph of the function u = x^a, we need to understand the behavior of the function based on the value of the exponent (a).

Since the value of a is positive (approximately 1.901307), the graph of the function will be an increasing curve. As x increases, u will also increase.

To sketch the graph, we can choose some x-values and calculate the corresponding u-values using the given exponent (a). Here are a few points:

- When x = 0, u = 0^1.901307 = 0. - When x = 1, u = 1^1.901307 = 1. - When x = 2, u = 2^1.901307 ≈ 3.364. - When x = 3, u = 3^1.901307 ≈ 7.590. - When x = 4, u = 4^1.901307 ≈ 16.970.

By plotting these points and connecting them with a smooth curve, we can sketch the graph of the function u = x^a.

Solution to Problem 9:

Unfortunately, the problem statement for problem 9 is not provided. Could you please provide the problem statement so that I can help you with the solution?

Solution to Problem 10:

Similarly, the problem statement for problem 10 is not provided. Please provide the problem statement so that I can assist you with the solution.