9. Alex and Samantha were given instructions to solve the numerical expression below. Are the instructions correct? Explain your reasoning.
6.5² + 8.5 ÷ 2² → Square 6.5. Add 8.5 to this number. Divide the sum by 2 and then square the quotient.
No, the instructions are not correct. According to the given instructions, the expression should be solved as follows:
1. Square 6.5: (6.5)² = 42.25
2. Add 8.5 to this number: 42.25 + 8.5 = 50.75
3. Divide the sum by 2: 50.75 ÷ 2 = 25.375
4. Square the quotient: (25.375)² = 644.765625
However, the actual expression is 6.5² + 8.5 ÷ 2².
In the actual expression, the order of operations should be followed, which is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
Using PEMDAS, we would solve the expression as follows:
1. Solve the exponent: 6.5² = 42.25
2. Solve the exponent: 2² = 4
3. Divide: 8.5 ÷ 4 = 2.125
4. Add: 42.25 + 2.125 = 44.375
So, the correct answer is 44.375.
The instructions given are not correct.
According to the given expression, 6.5² + 8.5 ÷ 2², the correct order of operations to be followed is as follows:
1. Start with the division operation: 8.5 ÷ 2².
This means we need to find the square of 2 first, which is 4.
Then, divide 8.5 by 4, which gives 2.125.
2. Next, perform the square operation on 6.5. The square of 6.5 is 42.25.
3. Finally, add the result of step 1 to the result of step 2.
42.25 + 2.125 = 44.375.
So, the correct expression, following the order of operations, is 44.375.
To determine whether the instructions are correct, we need to follow the steps and evaluate the expression.
Step 1: Square 6.5
When we square 6.5, we get 42.25.
Step 2: Add 8.5 to the result from Step 1
The sum of 42.25 and 8.5 is 50.75.
Step 3: Divide the sum by 2
When we divide 50.75 by 2, we get 25.375.
Step 4: Square the quotient
When we square 25.375, we get 644.051.
So, according to the given instructions, the numerical expression would be evaluated as 644.051.
However, it is important to note that the expression is ambiguous due to the lack of parentheses. The given expression 6.5² + 8.5 ÷ 2² can be interpreted in two ways:
1. (6.5²) + (8.5 ÷ 2²)
In this case, we first square 6.5 (which gives 42.25) and then divide 8.5 by 2² (which is 2). Finally, we add the results together to get 44.25.
2. (6.5² + 8.5) ÷ 2²
In this case, we first add 6.5 squared (42.25) and 8.5 (which gives 50.75). Then, we divide the sum by 2² (which is 4). Finally, we get 12.6875.
Without further clarification about the intended order of operations, it is impossible to determine which interpretation is the correct one. Therefore, the instructions are not clear and can lead to different results depending on the intended meaning.