Problem 54 - Algebra and Trigonometry

Question

Evaluate each expression. $$ \begin{array}{ll}{\text { (a) }|\sqrt{5}-5|} & {\text { (b) }|10-\pi|}\end{array} $$

Step-by-Step Solution

Answer

(a) 2.764, (b) 6.859.

Step 1: Understanding the Absolute Value Concept

Absolute value of a number is the non-negative value of the number without regard to its sign. For example, \(|x| = x\) if \(x \geq 0\) and \(|x| = -x\) if \(x < 0\).

Step 2: Calculating \(\sqrt{5}\)

The square root of 5 is approximately 2.236. This will be used in part (a) to calculate \( |\sqrt{5} - 5| \).

Step 3: Evaluating Part (a): \(|\sqrt{5} - 5|\)

Subtract \(5\) from \(\sqrt{5}\): \(2.236 - 5 = -2.764\). Calculate the absolute value, which is the positive value: \(|\sqrt{5} - 5| = 2.764\).

Step 4: Identifying the Value of \(\pi\)

The value of \(\pi\) is approximately 3.141. We will use this approximation to find \(|10 - \pi|\) in part (b).

Step 5: Evaluating Part (b): \(|10 - \pi|\)

Subtract \(\pi\) from \(10\): \(10 - 3.141 = 6.859\). Since the result is already positive, the absolute value is the same: \(|10 - \pi| = 6.859\).

Key Concepts

Square Root

The square root of a number is a value when multiplied by itself returns the original number. For instance, if you take the square root of 9, you'll get 3, because 3 x 3 equals 9. When dealing with non-perfect squares, like 5, the solutions are often expressed as decimals or left in a root form.
For example, the square root of 5 approximately equals 2.236, because 2.236 squared is very close to 5. It’s important in algebra to recognize when a number might not square perfectly and to use approximation intelligently.
When dealing with exercises involving square roots, it's helpful to remember the following:

• Simplify the expression before approximating.
• Check if the square root can stay in its root form for precision.
• Use a calculator to get accurate decimal values when needed.

Employing these strategies can help streamline the problem-solving process and improve accuracy when calculating square roots.

Pi

Pi, denoted as \( \pi \), is a significant constant in mathematics that represents the ratio of a circle's circumference to its diameter. This number is approximately equal to 3.14159, though it is commonly rounded to 3.141 in many calculations.
Pi is an irrational number, meaning it cannot be exactly expressed as a simple fraction, and its decimal form goes on forever without repeating. This property makes it particularly interesting and useful in various fields of both math and science.
Here are a few key points to consider when working with \( \pi \):

• Always use the most accurate approximation of \( \pi \) your problem allows.
• In more precise calculations, use as many decimal places as needed.
• Remember \( \pi \) is frequently involved in geometry, especially with formulas concerning circles.

Understanding \( \pi \) deeply is necessary for solving problems involving circular or spherical shapes.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, or division. They form the backbone of algebra and are used to represent relationships between quantities.
For example, in the expression \( x + 5 \), \( x \) is a variable that can take on different values, while 5 is a constant. The expression denotes all numbers that are 5 more than \( x \).
When working with algebraic expressions, some key aspects include:

• Simplifying expressions by combining like terms.
• Using the distributive property to expand expressions: \( a(b + c) = ab + ac \).
• Solving expressions for a variable, which means finding the value(s) that make the equation true.

Essentially, mastering algebraic expressions is foundational for solving equations and understanding more advanced mathematical concepts.