Problem 37 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{\frac{36}{25}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{6}{5} \)

Step 1: Identify the perfect squares

In the given expression \( \sqrt{\frac{36}{25}} \), both 36 and 25 are perfect squares. This means that we can take the square roots of both numbers.

Step 2: Take the square root of the numerator and the denominator

Square root of 36 is 6 and the square root of 25 is 5. Replace the square root of 36 by 6 and square root of 25 by 5 in the expression, we get \( \frac{6}{5} \).

Key Concepts

Perfect Squares

Understanding perfect squares is essential when it comes to simplifying expressions involving square roots. A perfect square is an integer that can be expressed as the product of another integer multiplied by itself. For example, 16 is a perfect square because it can be written as 4 times 4. In our exercise, both 36 and 25 are identified as perfect squares since 36 is 6 times 6, and 25 is 5 times 5.

When working with square roots, recognizing perfect squares allows for a straightforward simplification process, as you can replace the square root of a perfect square with its positive root. This is because the square root function undoes the squaring process: the square root of 36 (\( \text{which is } 6^{2} \text{)}\) will be 6, and similarly, the square root of 25 (\( \text{which is } 5^{2} \text{)}\) will be 5.

Square Root

The square root of a number is essentially a value that, when multiplied by itself, gives the original number. It answers the question: 'What number times itself equals this number?'. In our practice exercise, simplifying square roots involves finding the square root of the numerator (36) and the denominator (25) of the rational expression.

The number that produces the original perfect square through multiplication is the square root, like 6 is for 36, and 5 is for 25. This means we can substitute the square roots directly with these integers, immediately simplifying the expression without the need for long, drawn-out calculations.

Rational Expressions

A rational expression is a fraction in which both the numerator and the denominator are integers or polynomials. Our example deals with a rational expression because it presents a ratio of two integers, 36 and 25. When simplifying rational expressions that involve square roots, we aim to rationalize them by finding and extracting the square roots of perfect squares within the fraction.

In our given problem, once we have determined that both the numerator and the denominator are perfect squares, we proceed to find their square roots. By replacing the numerator and the denominator with their respective roots, the expression becomes a simplified rational expression, which is easier to interpret and work with in further mathematical operations or applications.