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Chapter 1: Problem 99

Evaluate \(\sqrt{a^{2}+b^{2}}\) when \(a=-3\) and \(b=4 .[\text { A. } 1]\)

Short Answer

Expert verified
The result of the evaluation of the mathematical expression \( \sqrt{a^{2}+b^{2}} \) when \( a=-3 \) and \( b=4 \) is 5.

Step by step solution

01

Substitute given values

Substitute the given values \( a=-3 \) and \( b=4 \) into the expression. The expression becomes \( \sqrt{(-3)^{2}+(4)^{2}} \).
02

Solve the exponentials

First, calculate the squares. In the brackets, \( (-3)^{2} \) becomes 9 and \( (4)^{2} \) becomes 16.
03

Prepare for the square root operation

The equation becomes \( \sqrt{9+16} \) after simplification.
04

Apply the square root operation

Next, perform the addition within the square-root leading to \( \sqrt{25} \). Now, take the square root of 25, leading to the result 5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root
The square root is a number that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, since 5 x 5 equals 25. When evaluating an expression like \(\sqrt{a^2 + b^2}\), first, you combine the values inside the square root, then find the square root of the result. In our exercise, after calculating \(a^2 + b^2 = 25\), we take \(\sqrt{25}\) to get 5. This is because 5 is the number that squares back to 25. Understanding this concept is key when working with both simple and complex expressions.
Exponentiation
Exponentiation involves raising a number to a power. For example, \((-3)^2\) means multiplying -3 by itself, resulting in 9. Similarly, \(4^2\) is 4 multiplied by 4, which equals 16. Exponentiation helps in simplifying expressions by transforming numbers into more manageable quantities. In our exercise, we used exponentiation to evaluate \((-3)^2 + 4^2\), simplifying it to \(9 + 16\). This step is crucial because it reduces the expression into sums of squares, setting up for easy simplification and square root calculation.
Simplification
Simplification in mathematics is all about reducing expressions to their simplest form. This involves performing operations like addition, subtraction, multiplication, and division. In the given exercise, simplifying \(9 + 16\) results in 25, which is a crucial step before applying operations like square root. Simplification helps make complex problems more manageable and paves the way for further operations, such as finding square roots or solving equations. By breaking down a problem into simpler steps, we can understand and solve it efficiently.

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Most popular questions from this chapter

Solve by completing the square or by using the quadratic formula. $$-x^{2}-7 x-1$$

Given the points \(P_{1}(-3,4)\) and \(P_{2}(2,-4),\) evaluate \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)

A manufacturer finds that the cost \(C\) in dollars of producing \(x\) items of a product is given by $$C(x)=(225+1.4 \sqrt{x})^{2}, \quad 100 \leq x \leq 1000$$ Complete the following table by evaluating \(C\) (to the nearest dollar) for the indicated numbers of items. $$\begin{array}{|c|c|c|c|c|c|}\hline x & 100 & 200 & 500 & 750 & 1000 \\ \hline C(x) & & & & & \\\\\hline\end{array}$$

Solve by completing the square or by using the quadratic formula. $$2 x^{2}+\sqrt{5} x-3=0$$

Use interval notation to express the solution set of each inequality. $$|x+3|>30$$
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