Problem 24 - Algebra 1: Concepts and Skills

Question

Write the equation in words. $$ \sqrt{\frac{1}{9}}=\frac{1}{3} $$

Step-by-Step Solution

Answer

The equation in words is: the square root of one divided by nine equals one-third.

Step 1: Identify the Squared Root

The first step focuses on interpreting the square root. The symbol \(\sqrt{}\) means 'square root of'. In our case, it translates to 'square root of 1 divided by 9'.

Step 2: Identify the Equal Sign

The equal sign in the context of mathematics is a symbol of equivalence; it means 'is equal to'.

Step 3: Identify the Fraction

The fraction should be interpreted as 'one-third'.

Key Concepts

Fraction

Fractions are a fundamental concept in mathematics. They represent a part of a whole or a division of quantities. A fraction consists of two numbers separated by a horizontal line. The top number is called the numerator, and it tells you how many parts you have. The bottom number, or the denominator, tells you into how many equal parts the whole is divided.
For example, in the fraction \( \frac{1}{9} \), the numerator is 1, and the denominator is 9. This fraction represents one part of a whole that is divided into nine equal parts. Another common example is \( \frac{1}{3} \), which means one part out of three equal parts. Fractions are used extensively to express division and ratios in a clear and concise manner.
Understanding fractions is key to solving problems involving percentages, ratios, and portions, making this concept a building block in mathematics education.

Equal Sign

The equal sign, denoted by \( = \), is a mathematical symbol used to indicate equality between two expressions. It represents a statement that the values or quantities on either side of the sign are exactly the same.
In the exercise \( \sqrt{\frac{1}{9}}=\frac{1}{3} \), the equal sign tells us that the square root of \( \frac{1}{9} \) is exactly \( \frac{1}{3} \). This is crucial in mathematical equations because it signifies balance. Just like a well-balanced scale, both sides of an equation hold all calculations equal.
This symbol is essential for understanding algebraic relationships, balancing equations, and verifying calculations across all fields of mathematics. In word problems or real-world applications, the equal sign helps clarify that two different expressions are indeed equivalent.

Mathematical Symbols

Mathematical symbols are a universal language in the field of mathematics, providing a way to communicate complex ideas succinctly. There are many common symbols such as \( +, -, \times, \div \) for basic operations. In the given exercise, it involves several important symbols:

• The square root symbol \( \sqrt{} \) indicates the operation of finding a number which, when multiplied by itself, gives the original number. This is a key concept in understanding radicals and simplifying mathematical expressions.
• The fraction bar in \( \frac{1}{9} \) and \( \frac{1}{3} \) means division. It shows how the parts are divided and simplifies equations for easier computation.
• The equal sign \( = \) indicates equivalency, showing that the expressions on either side of the sign represent the same quantity or value.

These symbols serve as the building blocks for more advanced topics, enabling a clear and concise depiction of mathematical operations and relationships across various problems and equations.