In the equation (43)_x = (y3)_8, both sides of the equation are written in different bases. The base of the first number is x and the base of the second number is 8.
To find the possible solutions for x and y, we need to find the range of values for each base.For the first number, (43)_x, the digits are 4 and 3. Since the base is x, the maximum digit value is x-1. Therefore, the maximum possible value for the first digit is (x-1)(x^1) = x. For the second digit, the maximum possible value is (x-1)(x^0) = x-1.For the second number, (y3)_8, the digits are y and 3. Since the base is 8, the maximum digit value is 7. Therefore, the maximum possible value for the first digit is 7(8^1) = 56. For the second digit, the maximum possible value is 7(8^0) = 7.
Now we compare the possible values for the digits. The maximum value for the first digit of (43)_x is x, and the maximum value for the first digit of (y3)_8 is 56. Since x cannot be greater than 56, we can conclude that x can take any value from 5 to 56. The second digit of (43)_x can take any value from 0 to x-1, and the second digit of (y3)_8 can take any value from 0 to 7. Therefore, there are infinitely many possible solutions for x and y.
(ii). In the equation √(224)r = (13)r, we need to find the value of the radix r. The equation states that the square root of (224)r is equal to the number (13)r. To find the value of r, we need to solve the equation.
First, let's convert the numbers to decimal form. The number (224)r can be written as 2r^2 + 2r + 4, and the number (13)r can be written as r + 3.
So, the equation becomes:
√(2r^2 + 2r + 4) = r + 3
To solve this equation, we can square both sides:
2r^2 + 2r + 4 = (r + 3)^2
2r^2 + 2r + 4 = r^2 + 6r + 9
Now, let's simplify the equation:
r^2 - 4r - 5 = 0
This is a quadratic equation. Factoring it, we get:
(r - 5)(r + 1) = 0
So, the possible values for r are r = 5 and r = -1. However, the radix r cannot be negative. Therefore, the only valid solution is r = 5.
(b) (i). If 137+276-435 equals 731+672, we can simplify both sides of the equation:
137 + 276 - 435 = 731 + 672
(137 + 276) - 435 = 1403
Therefore, the sum is 1403.
(ii). To find the base of the number system for the addition operation 24+14-41 to be true, we can examine the given numbers. The digits in the numbers are 2, 4, 1, and 4.
The base of the number system determines the range of possible digit values. To satisfy the equation, the highest possible digit value should be equal to or greater than the largest digit in the numbers.
In this case, the largest digit is 4. So, the base should be at least 5 to accommodate the digit 4. Therefore, the base of the number system for the addition operation 24+14-41 to be true is 5 or any higher base that can accommodate the largest digit, which is 4.