Problem 1 - College Algebra

Question

Evaluate each expression in Exercises \(1-6\) or indicate that the root is not a real number. $$\sqrt{36}$$

Step-by-Step Solution

Answer

The square root of \(36\) is \(6\).

Step 1 Identifying the Task

The task is to solve the square root of \(36\). This involves identifying the number that when multiplied by itself equals \(36\).

Step 2 Evaluate the Expression

To find the square root of \(36\), we need to identify the number that when multiplied by itself equals \(36\). It's known that \(6*6 = 36\), thus, \(\sqrt{36} = 6\).

Key Concepts

Real Numbers

Real numbers encompass all the numbers we encounter in everyday life, including whole numbers, decimals, and fractions. They can be either rational numbers, like 4 or 1.5, which can be written as fractions, or irrational numbers like \( \sqrt{2} \) or \( \pi \), which cannot be expressed as a simple fraction. Real numbers are crucial in mathematics as they help represent quantities comprehensively.

When dealing with square roots, like \( \sqrt{36} \), it's important to understand whether the result is a real number. The square root signifies a number which, when squared, gives the original number—in this case, 36. Here, \( \sqrt{36} = 6 \), a real number. This emphasizes that all non-negative numbers have real number square roots, although negative numbers do not have real number square roots as no real number squared will result in a negative value.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. The symbol for a square root is called a radical. It is written as \( \sqrt{} \). To simplify a radical expression like \( \sqrt{36} \, \), the goal is to find a number which, when multiplied by itself, results in the number inside the radical sign. This process is called evaluating or simplifying the radical expression.

To evaluate the radical expression \( \sqrt{36} \, \) we seek a number that multiplied by itself equals 36. From our knowledge or calculation, we know \( 6 imes 6 = 36 \), hence the radical \( \sqrt{36} \) simplifies to 6. Simplifying radicals is an essential skill in algebra as it helps transform complex expressions into more manageable forms.

Being comfortable with radicals allows students to solve various mathematical problems that involve different radical expressions and operations.

Algebraic Operations

Algebraic operations are fundamental processes used to manipulate mathematical expressions. They include addition, subtraction, multiplication, division, and operations with exponents and roots, such as square roots.

When performing the algebraic operation of taking a square root, like \( \sqrt{36} \), we are actually finding an equivalent number that works with multiplication. This step is crucial in solving many algebra equations as it helps identify values that satisfy an equation.

Operations involving radicals, such as addition \( (\sqrt{a} + \sqrt{b})\), are common in algebra and require understanding and practice. Managing radicals often involves simplifying, which means rewriting them in their simplest form. For instance, simplifying \( \sqrt{36} \) helps demonstrate that the operation yields a real number, 6, and reinforces the concept of equivalency in algebra, ensuring accurate solutions to equations.