CRITICAL THINKING :
Without grouping symbols,the expression 2.3³ + 4 has a value of 58.Insert grouping symbols in the expression 2.3³ + 4 to produce the following values :
1. 65
2. 220
3. 4374
4. 279,936
Please help me.
Show all how you solve it..
To put grouping symbols in the expression 2.3³ + 4 to produce the given values, we need to determine the order of operations within the expression.
1. To get a value of 65:
We can use the grouping symbols to emphasis the exponent calculation. We need to raise 2.3 to the power of 3 first, and then add 4.
So, the expression with grouping symbols would be: (2.3³) + 4
2. To get a value of 220:
We need to multiply 2.3 by itself three times (2.3 * 2.3 * 2.3), and then add 4.
So, the expression with grouping symbols would be: (2.3 * 2.3 * 2.3) + 4
3. To get a value of 4374:
We need to raise 2.3 to the power of 3 first, then multiply it by 4.
So, the expression with grouping symbols would be: (2.3³) * 4
4. To get a value of 279,936:
We need to raise 2.3 to the power of 3 first, multiply it by 4, and then raise the entire result to the power of 4.
So, the expression with grouping symbols would be: ((2.3³) * 4)⁴
To summarize, here are the expressions with grouping symbols:
1. (2.3³) + 4 = 65
2. (2.3 * 2.3 * 2.3) + 4 = 220
3. (2.3³) * 4 = 4374
4. ((2.3³) * 4)⁴ = 279,936
To find the correct placement of grouping symbols in the expression 2.3³ + 4 that would produce the given values, we need to understand the order of operations and the exponentiation rule.
Let's break down each value and solve them one by one:
1. 65:
Since the expression without grouping symbols has a value of 58, we need to increase the value by 7 to reach 65. This means the grouping symbols should be placed around the exponentiation operation.
The expression with grouping symbols would be: (2.3³) + 4.
This evaluates to: (2.3 × 2.3 × 2.3) + 4 = 12.167 + 4 = 16.167.
So, by placing the grouping symbols around the exponentiation operation, we get a value of 16.167, which is not equal to 65. Hence, this is not the correct solution.
2. 220:
To reach a value of 220, we need to increase the original value by 162. We can achieve this by placing the grouping symbols around the entire expression.
The expression with grouping symbols would be: (2.3³ + 4).
This evaluates to: (2.3 × 2.3 × 2.3) + 4 = 12.167 + 4 = 16.167.
So, by placing the grouping symbols around the entire expression, we still get a value of 16.167, which is not equal to 220. Therefore, this is also not the correct solution.
3. 4374:
To reach a value of 4374, we need to increase the original value by 4316. Again, we can achieve this by placing the grouping symbols around the entire expression.
The expression with grouping symbols would be: (2.3³ + 4).
This evaluates to: (2.3 × 2.3 × 2.3) + 4 = 12.167 + 4 = 16.167.
So, by placing the grouping symbols around the entire expression, we still get a value of 16.167, which is not equal to 4374. Therefore, this is not the correct solution.
4. 279,936:
To reach a value of 279,936, we need to increase the original value by 279,878, which is a much larger increase. This means we need to perform the exponentiation operation multiple times.
The expression with grouping symbols would be: (2.3)³ + 4.
This evaluates to: (2.3 × 2.3 × 2.3) + 4 = 12.167 + 4 = 16.167.
Since the grouping symbols are only around the exponentiation operation, we obtain the same value of 16.167 as before. Thus, this is not the correct solution either.
After analyzing all the possibilities, we can conclude that there is no placement of grouping symbols in the expression 2.3³ + 4 that would produce the given values.