Question

Evaluate each expression. Retain the proper number of significant digits in your answer. Roots Find each principal root without using your calculator. $$\sqrt{49}$$

Short answer

\sqrt{49} = 7

Step-by-step solution

- Identify and Understand the Principal Root

The principal root of a positive number is its positive square root. The symbol for the principal root is \( \sqrt{x} \) where x is a positive number. Since 49 is a positive number, you need to find the positive number that, when multiplied by itself, gives 49.

- Find the Square Root

In this case, the number that multiplies by itself to give 49 is 7, because \(7 \times 7 = 49\). Thus, \( \sqrt{49} = 7 \) is the principal root.

- Consider Significant Digits

The number 49 has two significant digits. However, since our answer is an exact number, there's no approximation or decimal, so we don't need to adjust the significant digits. The answer retains the proper number of significant digits without change.

Key concepts

Significant Digits

Significant digits, also known as significant figures, are the digits in a number that contribute to its accuracy. They are essential in measurements and calculations because they represent the precision of a value. When performing any mathematical operation, it’s vital to ensure that the result does not imply greater precision than the precision provided by the initial measurements. For instance, if you are told that a rectangle has a length of 5.30 meters and a width of 2.1 meters, you would report the area as 11.13 square meters, but if maintaining proper significant figures, you should round it to 11 square meters because the width was only given to two significant digits.

The rules to identify significant digits are as follows:

• Any non-zero digit is significant.
• Zeros between non-zero digits are significant.
• Leading zeros are not significant.
• Trailing zeros in a decimal number are significant.
• In a whole number, trailing zeros may or may not be significant depending on whether a decimal point is specified.

In the exercise example, the number 49 has two significant digits, indicating that the number is precise up to two places.

Square Root

The square root of a number is a value that, when multiplied by itself, yields the original number. Every non-negative real number has a unique non-negative square root, called the principal square root, and is denoted by \( \sqrt{x} \), where \( x \) is the number in question. If x is positive, then the square root represents a unique positive number which, when squared, returns x. For example, the principal square root of 9 is 3, because \( 3 \times 3 = 9 \). Conversely, the square root of 0 is 0 since \( 0 \times 0 = 0 \).

Finding square roots is a fundamental concept in mathematics that leads to more complex operations involving radicals and presents the basis for solving quadratic equations. Understanding square roots also allows us to comprehend geometric concepts such as the Pythagorean theorem and the areas of squares.

Radicals

Radicals refer to the symbols used to indicate roots. The most common radical is the square root symbol \( \sqrt{} \). However, radicals can also be used to represent other roots like cube roots \( \sqrt[3]{} \) or fourth roots \( \sqrt[4]{} \). The number under the radical sign is called the radicand, and radicals can encompass not only numbers but also variables and expressions.

The properties of radicals are crucial to their manipulation and simplification. Some basic ones include:

• \( \sqrt{x} \times \sqrt{y} = \sqrt{x \times y} \)
• \( \sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}} \)
• For any positive integer n, \( \sqrt[n]{x^n} = x \) (given x is non-negative).

Understanding how to work with radicals helps in simplifying expressions and solving equations, which is an integral part of algebra.

Square Root Calculation

The process of finding the square root of a number involves several methods, depending on the context and complexity of the number. For perfect squares, such as 49, the calculation is straightforward because the square root is an integer. In this situation, the process simply involves determining which integer value, when squared, equates to the given number. For 49, we intuitively know that 7 is the square root because \( 7 \times 7 = 49 \).

However, when dealing with numbers that are not perfect squares, one can use various methods such as prime factorization, long division, or even approximation methods like the Babylonian method, also known as the Hero's method. It's also worth noting that the advancement of technology has provided us with tools such as calculators and computers that can find square roots instantaneously. Notwithstanding, having a conceptual understanding of square root calculation is beneficial for learning more complex math topics and for situations where technology might not be available.