Question

Calculator Experiment Square the real number \(2 / \sqrt{5}\) and note that the radical is eliminated from the denominator. Is this equivalent to rationalizing the denominator? Why or why not?

Short answer

No, squaring the number \(2 / \sqrt{5}\) and eliminating the radical from the denominator is not equivalent to rationalizing the denominator. Though it results in a rational number, the original value of the number is changed, something that does not happen with rationalization.

Step-by-step solution

Squaring The Number

To find the square of the given real number \(2 / \sqrt{5}\), apply the formula \(a^2\), where 'a' is the number to be squared. Here, \(a = 2 / \sqrt{5}\), therefore, \((2 / \sqrt{5})^2= 4 / 5\).

Comparing the Results with Rationalizing the Denominator

Squaring the number has removed the radical from the denominator, we are now left with a rational number \(4 / 5\). However, in the process of removing the radical, the number itself was changed, whereas when rationalizing the denominator, the number itself does not change, only the type of number that makes up the denominator.

Formulating Conclusion

Although the act of squaring can remove a radical from the denominator, it is different from rationalizing the denominator. Rationalizing involves finding an equivalent fraction with a rational denominator while maintaining the original value of the fraction. Squaring the number changes the actual value of the original number.

Key concepts

Square Numbers

When we discuss square numbers, we're referring to numbers which result from multiplying a number by itself. This is expressed by the formula \(a^2\). For example, \(2^2\) equals 4. When dealing with fractions like \( \frac{2}{\sqrt{5}} \), the concept of square numbers changes slightly. We square the entire fraction rather than the numerator or denominator alone. This means the operation becomes \( \left( \frac{2}{\sqrt{5}} \right)^2 = \frac{2^2}{(\sqrt{5})^2} = \frac{4}{5} \).

In this process, the radical in the denominator of the fraction disappears because squaring a square root "undoes" the square root operation. Therefore, this concept is useful for simplifying fractions with radicals in the denominator when adjustments to the fraction's value are acceptable.

It's crucial to remember that squaring changes the original value of the number completely, as opposed to just changing its expression form. Hence, although useful, it serves a different purpose than merely simplifying the form of the original number.

Radicals in Algebra

Radicals are expressions that contain a square root, cube root, or any other root. They are common in algebraic expressions and equations. For example, \( \sqrt{5} \) is a radical that often appears in fraction denominators.

Working with radicals can complicate algebraic expressions, as they introduce irrational numbers. This is why eliminating radicals from denominators, known as rationalizing, is often performed. In our example, the denominator \( \sqrt{5} \) is made rational through squaring, although this changes the original value of the expression. In algebra, keeping the original expression's value while making its form more manageable is crucial.

Rationalizing typically involves multiplying the numerator and denominator by a conjugate or similar term to eliminate the radical, rather than altering the original number. This allows one to maintain the "essence" of the original fraction, ensuring calculations remain consistent in value and valid across algebraic operations.

Equivalent Fractions

Equivalent fractions are fractions that represent the same value or proportion, even though they may appear different. They are obtained by multiplying or dividing the numerator and denominator by the same number, maintaining equality.

In rationalizing the denominator, the goal is to transform the fraction into an equivalent one with a rational number in the denominator without changing its value. For instance, starting with \( \frac{2}{\sqrt{5}} \), we might want to rationalize by multiplying the top and bottom by \(\sqrt{5}\), obtaining \( \frac{2\sqrt{5}}{5} \). This new fraction is equivalent to the original because it represents the same value.

It's important to note that squaring to eliminate a radical, as we did in our initial example, changes the true value of the fraction, making it a different mathematical expression, thus not an equivalent fraction. Rationalizing aims to maintain the fraction's equivalency while offering clearer or more useful numeric representations in algebraic computations.