Problem 50 - Prealgebra

Question

Compute the exact square root. \(\sqrt{\frac{529}{16}}\)

Step-by-Step Solution

Answer

The exact square root is \(\frac{23}{4}\).

Step 1: Simplify the Expression Under the Square Root

The given expression is \(\sqrt{\frac{529}{16}}\). To simplify it, we first consider the fraction \(\frac{529}{16}\) separately. Since both the numerator and denominator are perfect squares, calculate the square root of each individually:

Step 2: Calculate the Square Root of the Numerator

The numerator is 529, which is a perfect square. To find the square root of 529, check known squares of integers. Since \(23^2 = 529\), the square root of 529 is 23.

Step 3: Calculate the Square Root of the Denominator

The denominator is 16, which is also a perfect square. The square root of 16 is 4, since \(4^2 = 16\).

Step 4: Combine Results

Now, combine the results from the previous steps. The expression becomes \(\frac{\sqrt{529}}{\sqrt{16}} = \frac{23}{4}\).

Step 5: Conclude the Simplified Square Root

Thus, the exact square root of \(\frac{529}{16}\) is \(\frac{23}{4}\).

Key Concepts

Fractions

A fraction is a mathematical expression representing a part of a whole. It is written as two numbers, one on top of the other, separated by a line. The top number is known as the numerator, and the bottom number is the denominator.
In our example, the fraction is \(\frac{529}{16}\). Here, 529 is the numerator, and 16 is the denominator.
Fractions can express divisions, ratios, or even quantities less than one.
Some common types of fractions are:

• Proper fractions: where the numerator is less than the denominator.
• Improper fractions: where the numerator is greater than or equal to the denominator.
• Mixed numbers: which combine a whole number and a proper fraction.

Understanding how to work with fractions is crucial in many areas of mathematics, including the calculation of square roots as used in this exercise.

Perfect squares

Perfect squares are integers that can be expressed as the square of another integer. For example, 16 is a perfect square because it equals \(4^2\), and similarly, 529 is a perfect square because it equals \(23^2\).
When dealing with square roots, especially inside fractions, recognizing perfect squares makes simplifying the expression much easier. This is because the square root of a perfect square is always an integer.
Some examples of perfect squares include:

• \(1^2=1\)
• \(2^2=4\)
• \(3^2=9\)
• \(10^2=100\)

Having a list of perfect squares in mind can save time when simplifying square roots in both single numbers and fractions.

Numerator and Denominator

The terms numerator and denominator are fundamental when working with fractions. The numerator (top number) indicates how many parts of the whole are being considered. The denominator (bottom number) tells us into how many parts the whole is divided.
In the fraction \(\frac{529}{16}\), 529 is the numerator and 16 is the denominator. Understanding these roles helps with operations such as simplifying fractions, finding common denominators, and working through mathematical concepts like square roots.
Here's how it applies to our task:

• To find the square root of a fraction, you find the square root of the numerator and the square root of the denominator separately.
• If both the numerator and denominator are perfect squares (as they are here), each root will be an integer, leading to an exact and simplified solution.

The ability to manipulate the numerator and denominator is essential for advanced mathematical topics.