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The absolute value of a number is its distance from zero on the number line, regardless of direction. In mathematical terms, the absolute value of a number \( x \) is denoted as \( |x| \). The concept is easy to grasp because it essentially captures the magnitude of a number without considering its sign.

• If \( x \) is positive, \( |x| = x \).
• If \( x \) is negative, \( |x| = -x \), which turns it positive.

Let's use the absolute value with a practical example from the exercise. In part (a), after computing \( 3.2^2 \) and \( \sqrt{3.15} \), we find the difference as 8.4652. Since this number is already positive, the absolute value remains \( |8.4652| = 8.4652 \). Hence, absolute values simplify calculations by ensuring we end up with non-negative results.

Square root approximation is a method used to estimate the square root of a number that isn't a perfect square. When we can't find an exact whole number square root, we approximate it using a calculator or estimation techniques. This process is invaluable for solving various mathematical problems.
When you approximate the square root using a calculator, you input the number and receive a decimal value. For example, in the exercise, \( \sqrt{3.15} \approx 1.7748 \). This estimation :

• helps in simplifying further calculations, and
• is used extensively in statistical and algebraic reasoning.

Understanding approximations allows you to decide how accurate you need to be for different problem requirements.

Power calculation involves determining the result of multiplying a number by itself a certain number of times. This is a fundamental operation in mathematics. It is denoted as \( x^n \), where \( x \) is the base, and \( n \) is the exponent, indicating how many times \( x \) is used as a factor.
In the exercise, \( 3.2^2 \) means multiplying 3.2 by itself, which calculates to 10.24. Here is a simple breakdown:

• Write down the base number twice.
• Multiply them together: \( 3.2 \times 3.2 = 10.24 \).

This results in what we call squaring, as the exponent is 2. Correctly calculating powers is critical for solving more complex mathematical operations, such as polynomials and complex equations.

The distance formula is used in mathematics to find the straight-line distance between two points in a plane. This formula is derived from the Pythagorean theorem and typically used in a Cartesian coordinate system.To apply the formula:

• Identify and define your points as \((x_1, y_1)\) and \((x_2, y_2)\).
• Plug these into the formula: \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).

In the exercise, we calculate the distances between two points: \((15.6, 4.3)\) and \((1.5, 5.4)\). Step by step, we find:- First, find the horizontal and vertical differences: \((15.6 - 1.5)^2 = 198.81\) and \((4.3 - 5.4)^2 = 1.21\)- Add these squares: \(198.81 + 1.21 = 200.02\)- Finally, take the square root to find the distance: \( \sqrt{200.02} \approx 14.1426 \)The distance formula helps us analyze geometric shapes, understand spatial relationships, and solve real-world problems with clarity.