Consider this expression: [tex]$\sqrt{a^3-7}+|b|$[/tex]
When [tex]$a=2$[/tex] and [tex]$b=-4$[/tex], what is the value of the expression?
Answer (2)
Substitute a = 2 and b = − 4 into the expression a 3 − 7 + ∣ b ∣ .
Calculate a 3 = 2 3 = 8 .
Calculate 8 − 7 = 1 = 1 .
Calculate ∣ − 4∣ = 4 , and then 1 + 4 = 5 . The final answer is 5 .
Explanation
Understanding the Problem We are given the expression a 3 − 7 + ∣ b ∣ and the values a = 2 and b = − 4 . Our goal is to find the value of the expression when we substitute these values for a and b .
Substituting the Values First, we substitute a = 2 and b = − 4 into the expression: 2 3 − 7 + ∣ − 4∣
Calculating 2 3 Next, we calculate 2 3 , which is 2 × 2 × 2 = 8 . So the expression becomes: 8 − 7 + ∣ − 4∣
Calculating 8 − 7 Now, we calculate 8 − 7 , which is 1 . The expression is now: 1 + ∣ − 4∣
Calculating Square Root and Absolute Value The square root of 1 is 1 , so 1 = 1 . The absolute value of − 4 is the distance from − 4 to 0 , which is 4 . Thus, ∣ − 4∣ = 4 . The expression simplifies to: 1 + 4
Adding the Values Finally, we add 1 and 4 to get 5 . Therefore, the value of the expression is 5 .
Examples
In physics, this type of expression could represent the magnitude of a force or energy, where 'a' and 'b' are variables affecting the overall value. For example, 'a' could be related to velocity and 'b' to a displacement. Evaluating such expressions helps engineers and physicists determine the net effect of these variables in a system, ensuring designs meet specific criteria or predicting system behavior under different conditions. Understanding how to substitute values and simplify expressions is crucial for solving real-world problems in science and engineering.
When substituting a = 2 and b = − 4 into the expression a 3 − 7 + ∣ b ∣ , we find that the value simplifies to 5 . This is calculated through various steps, including evaluating powers, square roots, and absolute values. Therefore, the final answer is 5 .
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