*Please explain on how to do each of these!
1. Let (x) be defined for all positive integer values of x as the product of all even factors of 4x. For example, (3)=12x6x4x2=576. What is the value of (5)? Someone says that it is f(5)= 20x10x2x4=1600 I DO NOT GET HOW THEY ARRIVED THAT ANSWER!
2. 2.3 ^5 log(is raised at the same level as 5 meaning not above 5) base 2.3 ^1
The answer is 1 but I do not get how?
3. If p and q are two different prime numbers greater than 2, and n=pq, how many positive factors, including 1 and n, does n have?
Do not get how the answer is 4!!!!????
4. How do you put this in a TI-83 graphing calc?
A= 9 -3 1
1 1 1
16 4 1
Matrix A is 3x3
X= x
y
z
Matrix X is a 3x1
B= -7
-3
21
Matrix B is 3x1
• Find AX=B
1. To find the value of (5), we need to calculate the product of all even factors of 4x, where x = 5.
Step 1: Identify the even factors of 4x. In this case, 4x refers to 4 multiplied by x.
The even factors of 4x are: 2, 4, 6, 8, ... 4x
Step 2: Substitute the value of x (= 5) into the even factors.
The even factors become: 2, 4, 6, 8, ... 20
Step 3: Calculate the product of these even factors.
Product = 2 * 4 * 6 * 8 * 10 * 12 * 14 * 16 * 18 * 20 = 16,777,216
Therefore, the value of (5) is 16,777,216.
The person you mentioned arrived at the answer of f(5) = 20 * 10 * 2 * 4 = 1,600, which is incorrect. They seem to have multiplied only a few of the even factors, rather than all of them.
2. The expression 2.3^5 * log(base 2.3) 2.3^1 simplifies to 1.
Step 1: Simplify the logarithm term within the expression.
Since the logarithm has the same base (2.3) as the number raised to the power, the logarithm simplifies to 1.
Step 2: Evaluate the expression.
2.3^5 * 1 = (2.3 * 2.3 * 2.3 * 2.3 * 2.3) * 1 = 643.6343... * 1 = 643.6343...
Therefore, the answer is 1.
3. To find the number of positive factors of n = pq, we need to consider the prime factors p and q.
Step 1: Identify the prime factors of n. In this case, n = pq.
Since p and q are different prime numbers greater than 2, let's say p > q.
The prime factors of n are p and q.
Step 2: Calculate the number of positive factors.
The number of positive factors of n = pq is given by the formula (a + 1) * (b + 1), where a and b are the exponents of the prime factors.
In this case, a = 1 (since the exponent of p is 1) and b = 1 (since the exponent of q is 1).
Number of positive factors = (1 + 1) * (1 + 1) = 2 * 2 = 4
Therefore, the number of positive factors of n = pq is 4.
4. To solve the equation AX = B using a TI-83 graphing calculator, follow these steps:
Step 1: Enter Matrix A into the calculator.
Press [2nd] [MATRIX] to access the matrix menu.
Choose "Edit" and press [ENTER].
Enter the values of Matrix A, which is a 3x3 matrix with entries 9, -3, 1, 1, 1, 1, 16, 4, 1.
Press [ENTER] to complete Matrix A.
Step 2: Enter Matrix B into the calculator.
Press [2nd] [MATRIX] to access the matrix menu.
Choose "Edit" and press [ENTER].
Enter the values of Matrix B, which is a 3x1 matrix with entries -7, -3, 21.
Press [ENTER] to complete Matrix B.
Step 3: Set up the equation AX = B.
Press [2nd] [MATRIX] to access the matrix menu.
Choose "Math," then go to "Matrix" and select "AB" (Matrix multiply) and press [ENTER].
Enter the name of Matrix A, "A," and press [ENTER].
Enter the name of Matrix X, "X," and press [ENTER].
Enter the name of Matrix B, "B," and press [ENTER].
Step 4: Solve for Matrix X.
To solve for X, press [2nd] [MATRIX] to access the matrix menu.
Choose "Math," then go to "Matrix" and select "Ax^-1B" (Matrix equation solver) and press [ENTER].
Enter the name of Matrix A, "A," and press [ENTER].
Enter the name of Matrix X, "X," and press [ENTER].
Enter the name of Matrix B, "B," and press [ENTER].
Press [ENTER] to solve for Matrix X.
The calculator will display the values of Matrix X, which represents the solutions for x, y, and z in the equation AX = B.
1. To find the value of (5), we need to compute the product of all even factors of 4x, where x=5 in this case. First, let's break down the calculation step by step:
Step 1: Find all the factors of 4x:
The factors of 4x are 1, 2, 4, 5, 10, 20, 4x, 2x, x.
Step 2: Identify the even factors:
The even factors are 2, 4, 10, 20, 4x, and 2x.
Step 3: Multiply the even factors:
Multiply all the identified even factors: 2 * 4 * 10 * 20 * 4x * 2x.
Now, let's simplify this expression:
2 * 4 * 10 * 20 * 4x * 2x = (2 * 2) * (2x * 4x) * (10 * 20)
= 4 * 8x^2 * 200
= 800x^2
Step 4: Substitute x=5:
Replace x with 5 in the simplified expression:
800 * (5^2) = 800 * 25 = 20,000
Therefore, the value of (5) is 20,000, not 1,600 as someone incorrectly claimed.
2. The expression 2.3^5 log base 2.3 ^1 can be simplified. Let's break down the calculation step by step:
Step 1: Evaluate 2.3^5 first:
Compute (2.3 * 2.3 * 2.3 * 2.3 * 2.3) = 643.6343
Step 2: Evaluate log base 2.3 ^1:
The logarithm base 2.3 of 1 is 0 because any number raised to the power of 0 is 1.
Step 3: Multiply both results:
643.6343 * 1 = 643.6343
Therefore, the answer is 643.6343, not 1 as you mentioned.
3. To determine the number of positive factors of n=pq, where p and q are different prime numbers greater than 2, we need to consider the prime factorization of n. Let's break it down step by step:
Step 1: Prime factorization of n:
Since p and q are prime numbers greater than 2, their only factors are 1 and themselves. Therefore, the prime factorization of n (pq) would be p * q.
Step 2: List all the factors of n:
To find all the factors of n, we need to consider all possible combinations of multiplying p and q. In this case, there are four combinations: 1, p, q, and pq. Hence, n has 4 positive factors: 1, p, q, and pq.
Therefore, the answer is 4, not anything else you mentioned.
4. To solve the equation AX=B using a TI-83 graphing calculator, follow these steps:
Step 1: Enter the values of matrix A:
Go to the matrix editor on your TI-83 calculator and enter the values of matrix A. Place the values in a 3x3 matrix.
A = 9 -3 1
1 1 1
16 4 1
Step 2: Enter the values of matrix B:
In the matrix editor, enter the values of matrix B. Place the values in a 3x1 matrix.
B = -7
-3
21
Step 3: Perform the calculation:
Using the matrix operations on the calculator, find the solution for X by solving AX=B. Go to the calculator's matrix operations, select the "rref" or "solve" function, and enter A and B as the matrices in the equation AX=B. Press the enter key, and the calculator will display the solution X.
The resulting value for X will depend on the calculator you are using, and it will be displayed in a matrix format.
That's how you can solve the equation AX=B using a TI-83 graphing calculator.