Problem 25 - Prealgebra

Question

Simplify each of the following expressions without using a calculator. $$\sqrt{\frac{16}{49}}$$

Step-by-Step Solution

Answer

The simplified form is \( \frac{4}{7} \).

Step 1: Understand the Problem

You are given the expression \( \sqrt{\frac{16}{49}} \) and you need to simplify it without a calculator. Simplification involves breaking it down using algebraic rules.

Step 2: Use the Property of Square Roots

The square root of a fraction \( \sqrt{\frac{a}{b}} \) can be split into \( \frac{\sqrt{a}}{\sqrt{b}} \). Apply this property to the given expression: \( \sqrt{\frac{16}{49}} = \frac{\sqrt{16}}{\sqrt{49}} \).

Step 3: Calculate the Square Roots

Find the square root of each term separately. \( \sqrt{16} = 4 \) because \( 4^2 = 16 \), and \( \sqrt{49} = 7 \) because \( 7^2 = 49 \).

Step 4: Simplify the Expression

Now simplify the expression by dividing the square roots, giving you \( \frac{4}{7} \).

Key Concepts

Understanding Algebraic Rules

When it comes to simplifying square roots, one must understand some essential algebraic rules and properties. These rules help us break down and simplify expressions more effectively and without the use of a calculator. A key algebraic property often used in simplification is the square root of a product. This rule states that the square root of a multiplication can be expressed as the multiplication of the individual square roots:

• \[ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \]

This property is quite useful because it allows us to handle complex expressions piece by piece.
Another important principle is that of coefficients and exponents. Understanding how exponents work can simplify algebraic expressions significantly. For example, know that the square root is the same as raising a number to the power of 1/2, which is very helpful when you consider complex expressions.
The algebraic world is consistent and predictable. By applying these rules consistently, we ensure that the expressions remain mathematically equivalent after simplification.

Square Root of a Fraction

The property of square roots involving fractions is incredibly handy during simplification exercises. It states that the square root of a fraction equals the square root of the numerator divided by the square root of the denominator:

• \[ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \]

This property allows us to work with simpler numbers, as you can calculate the square root of the top and bottom numbers separately.
Take for example, \( \sqrt{\frac{16}{49}} \). Applying this rule simplifies our task as we can find separate roots of 16 and 49 easily, which will then be placed over each other to give the answer.
By using this strategy, you can break down seemingly complex problems into smaller, more manageable problems.

The Simplification Process

Simplifying expressions, particularly those involving square roots, follows a logical series of steps. Begin by applying known algebraic rules and properties, like the square root of a fraction rule previously discussed. Once you've split the expression, calculate the square roots individually.
For instance, in our example \( \sqrt{\frac{16}{49}} \), after applying the property yields \( \frac{\sqrt{16}}{\sqrt{49}} \). Calculate these separately:

• The square root of 16 is 4 because \( 4^2 = 16 \).
• The square root of 49 is 7 because \( 7^2 = 49 \).

Now, by dividing these results, you achieve a simplified expression: \( \frac{4}{7} \).
Always check your work at each stage to ensure no steps were skipped and every transformation maintains the expression's value.
Simplification is like solving a puzzle where the right pieces have to fit the right places to achieve the final, simpler result.