Problem 6 - Elementary Algebra

Question

Name both square roots of each of the following numbers. $$ \frac{1}{4} $$

Step-by-Step Solution

Answer

Answer: The two square roots of the given number \(\frac{1}{4}\) are \(\pm\frac{1}{2}\).

Step 1: Identify the given number

The given number is a fraction: $$ \frac{1}{4} $$

Step 2: Find the square roots of the numerator and denominator separately

To find the square roots of the fraction, first find the square roots of the numerator and the denominator: $$ \sqrt{1}=1 $$ and $$ \sqrt{4}=2 $$

Step 3: Form the positive and negative square roots of the given fraction

Now, form the positive and negative square roots of the given fraction by dividing the square root of the numerator by the square root of the denominator: $$ \sqrt{\frac{1}{4}}=\frac{\sqrt{1}}{\sqrt{4}}=\frac{1}{2} $$ and $$ -\sqrt{\frac{1}{4}}=-\frac{\sqrt{1}}{\sqrt{4}}=-\frac{1}{2} $$

Step 4: Write down the two square roots of the given number

The two square roots of the given number, \(\frac{1}{4}\), are: $$ \pm\frac{1}{2} $$

Key Concepts

Simplifying Square Roots

The process of simplifying square roots involves finding an equivalent simpler or more explicit form of a square root without altering its value. To start with, it's crucial to understand that a square root of a number is a value that, when multiplied by itself, gives the original number. For fractions, this means we need to consider the square root of both the numerator and the denominator separately.

For example, if we have the fraction \(\frac{1}{4}\), simplifying the square root of this fraction entails calculating the square root of the numerator and the denominator. The square root of 1 is 1, and the square root of 4 is 2. Thus, the simplified square root of the fraction \(\frac{1}{4}\) is \(\frac{1}{2}\). Remember that every positive number has two square roots, a positive and a negative one, so it's essential to consider both when tackling problems involving square roots of fractions.

Radicals in Algebra

The term 'radicals' in algebra refers to expressions that contain a root, such as a square root, cube root, and so on. When dealing with radicals, it's important to know the rules for simplifying these expressions. A radical can often be simplified if the number under the root sign can be divided by a perfect square.

When simplifying radicals involving fractions like \(\sqrt{\frac{1}{4}}\), you simplify the roots of the numerator and the denominator as separate entities before combining them. Algebraically, this principle is reflected in the property \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\), where \(a\) and \(b\) are non-negative numbers. This property is useful because it can turn a radical of a fraction into a simpler form, and sometimes into an arithmetic operation of non-radical numbers.

Fractional Exponents

Fractional exponents are an alternative notation for expressing powers and roots in algebra. A fractional exponent consists of a numerator and a denominator: the numerator represents the power, and the denominator represents the root. This can be especially handy when dealing with square roots of fractions.

The expression \(x^{\frac{1}{2}}\) is equivalent to the square root of \(x\). So, when you see a fractional exponent with a 2 in the denominator, like \(\left(\frac{1}{4}\right)^{\frac{1}{2}}\), it's asking for the square root of \(\frac{1}{4}\), which as we've seen, is \(\frac{1}{2}\). It's important to be comfortable with switching between radical notation and fractional exponents because they are different ways of expressing the same mathematical reality. They are interchangeable, and understanding both allows for greater flexibility in problem-solving.