Problem 39 - Fundamentals of Mathematics

Question

Use the order of operations to determine each value. \(\sqrt{16+9}\)

Step-by-Step Solution

Answer

5

Step 1: Simplify the expression inside the square root

Start by adding the numbers inside the square root. The expression is \(16 + 9\). Calculate \(16 + 9\) to get \(25\).

Step 2: Apply the square root operation

Now find the square root of the result from Step 1. The square root of \(25\) is \(5\).

Key Concepts

Understanding the Square Root

The square root is a fundamental concept in mathematics that relates to finding a number which, when multiplied by itself, gives the original number. For example, \( ext{the square root of 25 is represented as} \, \\sqrt{25}\), \(because 5 times 5 equals 25.\) Learning how to work with square roots is essential in solving various mathematical problems. You'll often see a radical sign (√) used to denote a square root. When solving equations with square roots, the goal is to work towards making the expression under the radical sign as simple as possible.
A couple of important tips to remember about square roots:

• Square roots only apply to non-negative numbers in basic math operations.
• The square root of 0 is 0, as 0 times 0 equals 0.
• Whenever you see \(\sqrt{a + b}\), you need to add \({a}\) and \({b}\) first before applying the square root.

If you encounter a square root in a mathematical problem, taking it step by step usually helps clarify the path to a solution.

Adding Numbers

Adding numbers is one of the first and most essential operations you learn in mathematics. When you add, you are simply finding the total or the sum of two or more numbers. In our example, \(16 + 9\), \(the sum is 25.\)
It is important to start by aligning numbers correctly, particularly when working with larger numbers comprising multiple digits. Tools like mental math techniques or calculators can assist, yet understanding the basics is crucial for building a solid mathematical foundation.

• Add counting numbers naturally from left to right or from right to left.
• Remember to carry over to the next highest place value if a sum exceeds 9 in any digit place.
• Retention of basic addition facts can greatly enhance speed and accuracy in calculations.

Adding numbers is not only useful in solving exercises like this but also a vital skill across numerous real-life applications like budgeting and resource management.

Demystifying Mathematical Expressions

Mathematical expressions involve a combination of numbers, operations, and sometimes variables to convey a specific idea or quantity. They are the language of mathematics, enabling us to illustrate problem situations and solutions concisely.
In the original exercise, \(\sqrt{16+9}\), \(the expression tells us what steps to perform (add first, then apply the square root)\).
Key points about mathematical expressions include:

• Always follow the order of operations, typically remembered as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).
• Simplifying inside parentheses (or in our case, under a square root) comes before applying the operation outside.
• Expressions don't have an equal sign, distinguishing them from equations, which show equality.

Successfully working through mathematical expressions involves recognizing and applying arithmetic rules appropriately. This helps in accurately interpreting and solving complex problems, paving the way for advanced mathematical learning.