Problem 48 - Prealgebra

Question

Use the following information. Squaring a number and finding the square root of a number are inverse operations. That is, one operation undoes the other operation. Use inverse operations to evaluate each expression. $$(\sqrt{169})^{2}$$

Step-by-Step Solution

Answer

The expression \((\sqrt{169})^2\) evaluates to 169.

Step 1: Understanding the Expression

The expression given is \((\sqrt{169})^2\). It involves taking the square root of 169 and then squaring the result.

Step 2: Evaluating the Square Root

First, we find the square root of 169. The square root of 169 is 13, because \(13 \times 13 = 169\). So, \(\sqrt{169} = 13\).

Step 3: Applying the Square of the Result

Next, we square the result we found from the previous step. Since we found that \(\sqrt{169} = 13\), we now calculate \(13^2\).

Step 4: Simplifying the Expression

When we square 13, we compute \(13 \times 13 = 169\). Thus, \((\sqrt{169})^2 = 169\).

Key Concepts

Understanding Square Roots

A square root is a mathematical operation that finds a number which, when multiplied by itself, gives the original number. If you think of a square shape in geometry, the square root is like the length of one side of that square.
For example, when we take the square root of 169, we are looking for a number that, when multiplied by itself, equals 169. This number is 13, as shown in the solution, since \[13 \times 13 = 169.\]

• The operation is represented using the radical symbol, \(\sqrt{}\).
• Easy tip: 169 is a perfect square because its square root is an integer.

Square roots are handy in reversing the effect of squaring numbers, making it an important inverse operation in mathematics. The square root helps to simplify expressions and evaluate equations in various fields, from algebra to real-world applications like calculating areas or understanding wave functions.

The Concept of Squaring Numbers

Squaring a number means multiplying the number by itself. This operation is represented by a small 2 above the number, known as an exponent. So, when we see an expression like \[13^2,\]it means 13 multiplied by 13.
This results in 169, as demonstrated in the exercise. Squaring results in perfect squares when we start with whole numbers, such as

• 2 squared equals 4,
• 4 squared equals 16,
• 5 squared equals 25,

Squaring has many applications:- It is essential in calculating areas, especially for square shapes.- Used in statistics, like calculating variance.Understanding squaring is crucial as it often appears in algebraic expressions and equations, feeling relevant beyond basic mathematics. For example, any expression can involve squared terms, influencing how we graph and interpret results.

Delving into Mathematical Expressions

Mathematical expressions can be simple or quite complex and are essentially combinations of numbers, variables, and operation symbols. They don't contain an equal sign, unlike equations. For instance, the expression presented \[(\sqrt{169})^2\]is made of a square root and squaring operation, bringing together a dynamic interaction of inverse operations in a neat format.

Understanding expressions helps in:

• Identifying operations: Recognize what's being performed on the numbers (e.g., addition, subtraction, squaring, etc.).
• Simplifying problems: Break a complex problem into smaller, manageable parts.
• Evaluating results: Determine the effectiveness of operations used and checking work.

Mathematical expressions allow you to translate real-world scenarios into calculable and interpretative formats. Mastery of evaluating expressions paves the way for more advanced mathematics, such as calculus and beyond, where these foundational skills are key to understanding change and integration, whether in economics, engineering, or sciences.