Problem 76 - A Graphical Approach to College Algebra

Question

Approximate each expression to the nearest hundredth. $$1-\frac{4.5}{3-\sqrt{2}}$$

Step-by-Step Solution

Answer

The approximate value is -1.83.

Step 1: Simplify Inside the Denominator

First, simplify the expression inside the denominator. We have the square root:\[3 - \, \sqrt{2}\]Calculate \(\sqrt{2} \approx 1.41\). Thus, the expression becomes:\[3 - 1.41 = 1.59\]

Step 2: Divide the Numerator by the Simplified Denominator

Next, divide 4.5 by the simplified denominator 1.59:\[\frac{4.5}{1.59} \approx 2.83\]

Step 3: Subtract the Result from 1

Finally, subtract the result from 1:\[1 - 2.83 = -1.83\]

Step 4: Round to the Nearest Hundredth

Round \(-1.83\) to the nearest hundredth, which remains \(-1.83\), since it is already at the required precision.

Key Concepts

Square Root

The concept of a square root is fundamental in algebra and helps simplify expressions involving powers. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2, because 2 times 2 equals 4. This idea applies to any non-negative number.

When working with square roots of non-perfect squares, such as 2, we approximate the square root. The square root of 2 cannot be expressed as a simple fraction and is approximately 1.41. This approximation is useful in calculations where an exact value isn't necessary, allowing quicker and more accessible results for practical use cases.

• When simplifying expressions, estimate square roots to make calculations manageable.
• Use a calculator if precision is crucial beyond a rough approximation.

Having a good estimate of square roots helps in simplifying expressions by reducing complexity, especially when dealing with algebraic fractions as seen in exercises.

Simplifying Expressions

Simplifying expressions is a key algebraic skill that involves reducing them to their most straightforward form. This process helps in making calculations easier to perform and understand. An expression like \(1 - \frac{4.5}{3 - \sqrt{2}}\) can look challenging, but by breaking it down step by step, it becomes manageable.

Firstly, identify components in the expression that can be simplified individually. For instance, in the denominator \(3 - \sqrt{2}\), calculate the square root and then perform the subtraction to simplify it to 1.59.

Once simplified, divide the numerator 4.5 by the result. Continue simplifying by performing basic arithmetic, as in dividing 4.5 by 1.59 to get approximately 2.83.

• Always simplify within brackets or the denominator before handling the entire expression.
• Break down complex operations into smaller, easier steps.
• Check your calculations to avoid errors in the final expression.

By simplifying, you make solving algebraic problems less intimidating and more methodical.

Rounding Numbers

Rounding numbers is an essential skill, particularly when working with decimals in practical applications or to meet specific precision requirements. Rounding involves adjusting a number to a near approximation, facilitating easier communication and simpler math operations.

The rule of rounding depends on the numbers you are dealing with. If you are rounding to the nearest hundredth, observe the third decimal place:

• If it's 5 or greater, round the last significant digit up.
• If it's less than 5, keep the digit unchanged.

In the context of the exercise, the number -1.83 was already rounded to the nearest hundredth. If the result had been, say, -1.832 instead, it would have been rounded to -1.83, because the third digit (2) is less than 5.

Rounding is frequently used when precise numbers aren't necessary, allowing for easier interpretation of data and faster calculations, especially in estimating or approximations.