Question

Complete the statement with \(<,=\), or \(\geq$$5 \quad \sqrt{3^{2}+4^{2}}\)

Short answer

The symbol that completes the statement is '='. Thus, the completed statement is '5 = \( \sqrt{3^{2}+4^{2}}\)'

Step-by-step solution

Calculate the expression under the square root

Firstly, you need to apply the order of operations - PEMDAS/BODMAS to calculate the expression inside the square root - which is \(3^{2}+4^{2}\). Which equals to \(9+16=25\).

Square root the result

Next, calculate the square root of the outcome (25) from step 1. The square root of \(25\) is \(5\). Hence, \(\sqrt{3^{2}+4^{2}}\) equals \(5\).

Compare values

Finally, compare the calculated root value (5) with the left-hand side of the inequality (5). \nIt can be noted that '5' is equal to '5'. Hence, the symbol that completes the statement is '='.

Complete the statement

As the two values are equal, the final statement becomes: '5 = \(\sqrt{3^{2}+4^{2}}\)'.

Key concepts

Order of Operations

Understanding the order of operations is essential when solving mathematical problems. The order dictates the correct sequence in which to solve a complex equation. It is often remembered by the acronym PEMDAS or BODMAS, standing for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

For example, consider the expression \(3^{2}+4^{2}\) from our exercise. According to the order of operations, we first calculate the exponents before proceeding to addition. Thus, \(3^{2}\) and \(4^{2}\) are calculated as 9 and 16, respectively. Only after these operations are completed do we add the two results together to get 25. Failing to follow this order could result in an incorrect solution. Therefore, remember to always perform calculations in the right order to ensure accuracy.

Square Root Calculation

The square root is a crucial mathematical function, especially when working with geometric and algebraic problems. The square root of a number is a value that, when multiplied by itself, gives the original number. Calculating the square root of a number involves finding this value.

Using the exercise as an example, we found that the expression inside the square root, after applying the correct order of operations, is 25. To find the square root of 25, we ask ourselves, 'What number multiplied by itself equals 25?' The answer is 5 since \(5 \times 5 = 25\). We represent the square root of 25 as \(\sqrt{25}\), and it simplifies to 5. Understanding how to calculate square roots is essential for solving a variety of mathematical problems, including the one in our exercise.

Comparing Numbers

A basic skill in mathematics is comparing numbers to determine their relationship using inequality symbols such as <, =, and ≥. When comparing numbers, our goal is to ascertain whether a number is less than, greater than, or equal to another number.

In our exercise, after we calculated \(\sqrt{3^{2}+4^{2}}\) and found it to be 5, we needed to compare this value to another 5. Since the two numbers are identical, we conclude that they are equal. This comparison is straightforward since both values are the same. However, when dealing with different numbers or variables, we must carefully evaluate their size or value to choose the correct inequality symbol, ensuring our mathematical statements are accurate.