Problem 38 - Elementary Algebra

Question

For the following problems, find the two square roots of the given number. $$ \frac{1}{49} $$

Step-by-Step Solution

Answer

Answer: The two square roots of $\frac{1}{49}$ are $\frac{1}{7}$ and $-\frac{1}{7}$.

Step 1: Write the expression as a square root of a fraction

First, we need to rewrite the given expression as a square root of a fraction: $$ \sqrt{\frac{1}{49}} $$

Step 2: Find the square roots of the numerator and the denominator

Next, we need to find the square roots of both the numerator (1) and the denominator (49): $$ \sqrt{1} = \pm 1 $$ $$ \sqrt{49} = \pm 7 $$

Step 3: Simplify the expression

Now, we can divide the square roots of the numerator and the denominator: $$ \frac{\pm 1}{\pm 7} = \pm \frac{1}{7} $$

Step 4: Write the two square roots

Finally, we can write the two square roots of the given fraction: $$ \sqrt{\frac{1}{49}} = \frac{1}{7} \text{ and } -\frac{1}{7} $$

Key Concepts

Fractions in Algebra

In algebra, fractions often appear in expressions, equations, and functions. A fraction is composed of a numerator, which is the top part, and a denominator, which is the bottom part. They represent a ratio between two numbers.
Fractions are essential for expressing parts of a whole or a division of quantities.
In algebra, you frequently manipulate fractions to simplify expressions or solve equations.

• When working with fractions, always consider the relationship between the numerator and the denominator.
• Fractions can be combined with operations such as addition, subtraction, multiplication, and division.
• It's crucial to maintain an equal balance between the numerator and the denominator to preserve the fraction's value.

Understanding how to work with fractions in algebra is foundational, as it allows students to build on this knowledge with more advanced mathematical concepts.

Simplifying Expressions

Simplifying expressions is a key skill in algebra that involves reducing expressions to their simplest form. This includes combining like terms, reducing fractions, and eliminating unnecessary operations.
The process of simplification makes algebraic expressions easier to read and use for further calculations.

• To simplify a fraction, find the greatest common factor (GCF) of the numerator and the denominator and divide both by this factor.
• For expressions with square roots, simplify by taking the square root of both the numerator and the denominator separately, as shown in the problem solution.
• Use the property: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \) to simplify fractions under a square root.

Simplification leads to clearer and more manageable algebraic expressions, making calculations more straightforward.

Numerators and Denominators

The numerator and the denominator are two critical components of a fraction. Understanding their roles is essential when working with fractions in algebra.

The numerator, located above the fraction bar, indicates the number of equal parts being considered, while the denominator, below the bar, shows the total number of equal parts in a whole.

• The numerator can be any real number, integer, or algebraic expression.
• The denominator cannot be zero, as division by zero is undefined in mathematics.
• Simplifying a fraction involves dividing both the numerator and the denominator by their greatest common divisor (GCD) to achieve the simplest form.

By focusing on numerators and denominators, one gains a deeper understanding of how fractions function within expressions and equations.