Problem 19 - Algebra 2

Question

Simplify. $$ \sqrt{\frac{1}{16}} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{1}{4} \).

Step 1: Recognize the Expression

The expression given is \( \sqrt{\frac{1}{16}} \). Our goal is to simplify it, which means finding its simplest form.

Step 2: Apply the Square Root to the Fraction

The square root of a fraction \( \frac{a}{b} \) is calculated as \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \). Apply this to the expression to get \( \frac{\sqrt{1}}{\sqrt{16}} \).

Step 3: Calculate the Square Root of the Numerator

The square root of 1 is 1, as \( 1 \times 1 = 1 \). Thus, \( \sqrt{1} = 1 \).

Step 4: Calculate the Square Root of the Denominator

The square root of 16 is 4, as \( 4 \times 4 = 16 \). Therefore, \( \sqrt{16} = 4 \).

Step 5: Simplify the Fraction

Substitute back to see that \( \frac{\sqrt{1}}{\sqrt{16}} = \frac{1}{4} \). This is the simplest form of the original expression.

Key Concepts

Understanding Square Roots

Square roots are a mathematical concept used to find a number that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3 because when you multiply 3 by itself, you get 9. In symbolic terms, \( \sqrt{9} = 3 \). With square roots, the aim is to determine what number can be multiplied by itself to reach the number under the square root symbol, known as the radicand.
In the exercise, we had \( \sqrt{\frac{1}{16}} \), which entails finding numbers that when squared, reproduce 1 and 16 respectively. Recognizing this helps us handle more complex square root calculations efficiently.

Getting to Know Fractions

Fractions represent parts of a whole and are a fundamental part of arithmetic. A fraction consists of two parts: the numerator, which is the top number, and the denominator, which is the bottom number. For example, in the fraction \( \frac{1}{16} \), 1 is the numerator and 16 is the denominator. This fraction tells us that we have 1 part of something that is divided into 16 equal parts.
When working with fractions, remember:

• The fraction tells you how many equal parts there are (denominator) and how many parts are being considered (numerator).
• You can perform operations such as addition, subtraction, multiplication, and division with fractions when managed correctly.

Fractions often appear in various mathematical problems and understanding them is essential for working with other mathematical concepts, such as simplifying expressions.

Simplifying Fractions the Easy Way

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no common factor other than 1. To simplify a fraction, find the greatest common divisor (GCD) of the numerator and denominator and divide both by this number.
In the exercise, after finding the square roots, we simplified \( \frac{1}{4} \) by checking if there is a common factor. Here, 1 and 4 have no common factors other than 1, meaning \( \frac{1}{4} \) is already in its simplest form.

• Simplifying helps in making calculations easier and results more understandable.
• Always check if both the numerator and denominator can be divided by the same number until it cannot be simplified anymore.

By mastering simplification, you will sharpen your skills in handling any fraction-related problems you encounter in math.