Problem 43 - Prealgebra

Question

Compute the exact square root. \(\sqrt{\frac{529}{121}}\)

Step-by-Step Solution

Answer

The exact square root is \(\frac{23}{11}\).

Step 1: Simplify the Square Root Expression

Start by simplifying the expression under the square root. We have \(\frac{529}{121}\). The square root of a fraction \(\frac{a}{b}\) can be expressed as \(\frac{\sqrt{a}}{\sqrt{b}}\). Hence, \(\sqrt{\frac{529}{121}} = \frac{\sqrt{529}}{\sqrt{121}}\).

Step 2: Compute the Square Root of the Numerator

Evaluate the square root of the numerator, which is \(\sqrt{529}\). Recognizing 529 as a perfect square, we find \(\sqrt{529} = 23\) since \(23 \times 23 = 529\).

Step 3: Compute the Square Root of the Denominator

Next, find the square root of 121. Since 121 is a perfect square, \(\sqrt{121} = 11\) because \(11 \times 11 = 121\).

Step 4: Form the Resulting Fraction

Now, substitute the calculated square roots back into the expression: \(\frac{\sqrt{529}}{\sqrt{121}} = \frac{23}{11}\). Therefore, the exact square root of \(\frac{529}{121}\) is \(\frac{23}{11}\).

Key Concepts

Understanding Perfect Squares

To appreciate the idea of perfect squares, imagine a square shape where all sides are equal. A perfect square is essentially a number that can be expressed as the product of an integer multiplied by itself.
For example, 529 and 121 in the exercise are perfect squares because:- 529 can be obtained from multiplying 23 by itself: \(23 \times 23 = 529\) - 121 can be obtained from multiplying 11 by itself: \(11 \times 11 = 121\)

• Identifying perfect squares helps simplify complex problems, especially involving square roots.
• Recognizing numbers as perfect squares makes extracting their square roots straightforward, avoiding complex calculations.

Once a number is identified as a perfect square, one can directly determine its square root, which greatly eases the computations.

Mastering Fraction Simplification

Fraction simplification is crucial when working with mathematical expressions involving fractions. It allows expressions to become more manageable and often leads to understanding the problem better.
Simplification involves reducing the fraction to its smallest equivalent form. In this exercise, simplifying the expression \(\sqrt{\frac{529}{121}}\) is achieved by rearranging it into the square root of its numerator divided by its denominator: \(\frac{\sqrt{529}}{\sqrt{121}}\).

• By simplifying, we can focus on calculating square roots separately, which splits a complex problem into simpler tasks.
• Separating out these operations allows for efficient computation and reveals approximate numerical relationships.

More broadly, simplification aids in achieving the most reduced form that is often required in mathematical proofs and during practical computations.

Numerator and Denominator: Key Elements of Fractions

When dealing with fractions, understanding the roles of the numerator and denominator is critical. Each part of a fraction plays a distinct role:
- **Numerator**: The top number of a fraction, representing how many parts out of the whole are under consideration. In our example, 529 is the numerator. - **Denominator**: The bottom number of a fraction, indicating into how many parts the whole is divided. Here, 121 is the denominator.

• Knowing how the numerator and denominator relate is important for division operations, helping you manage computations in fraction form.
• The square root of a fraction, as simplified by separating roots for the numerator and denominator, manifests this well.

Access to this understanding assists in grasping both arithmetic and algebraic manipulations of fractions, easing the computations and improving accuracy.