Problem 17
Question
Use the quotient rule to simplify the expressions in Exercises \(17-26 .\) Assume that \(x>0\) $$\sqrt{\frac{1}{81}}$$
Step-by-Step Solution
Verified Answer
The simplified form of \(\sqrt{\frac{1}{81}}\) is \(\frac{1}{9}\).
1Step 1: Identify the square root
Our task is to simplify the expression \(\sqrt{\frac{1}{81}}\). Begin by observing that there is a square root covering the fraction.
2Step 2: Apply square root to numerator and denominator respectively
Because the square root of a fraction \(\sqrt{\frac{a}{b}}\) is equal to \(\frac{\sqrt{a}}{\sqrt{b}}\), we can apply the square root to the numerator (1) and denominator (81) separately, which becomes \(\frac{\sqrt{1}}{\sqrt{81}}\).
3Step 3: Simplify the square root
Calculate these square roots. The square root of 1 is 1, and the square root of 81 is 9. Therefore, our expression simplifies to \(\frac{1}{9}\).
Key Concepts
Simplifying Square RootsSquare Root of a FractionRadical Expressions
Simplifying Square Roots
To simplify square roots effectively, one must understand that the square root of a number is a value which, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 times 5 equals 25. Simplifying a square root involves finding the prime factors of the number and identifying any pairs of identical factors, as pairs can be taken out of the square root as their single base.
However, when the square root involves a perfect square (like 1, 4, 9, 16, etc.), simplification is straightforward since they have whole number square roots. Indeed, the square root of a perfect square results in a simplified whole number without the radical sign. In our exercise, since 1 and 81 are both perfect squares, their square roots are easily found to be 1 and 9, respectively.
However, when the square root involves a perfect square (like 1, 4, 9, 16, etc.), simplification is straightforward since they have whole number square roots. Indeed, the square root of a perfect square results in a simplified whole number without the radical sign. In our exercise, since 1 and 81 are both perfect squares, their square roots are easily found to be 1 and 9, respectively.
Square Root of a Fraction
When addressing the square root of a fraction, the process involves applying the square root to the numerator and the denominator separately because of the mathematical rule \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\). This rule is particularly helpful when dealing with fractions that have perfect squares in the numerator and the denominator.
By applying the square root to both the top and bottom numbers individually, they can each be simplified if possible, and then the results form a new, often simpler, fraction. In our exercise, applying the rule to \(\frac{1}{81}\), we evaluate \(\sqrt{1}\) and \(\sqrt{81}\) separately to obtain \(\frac{1}{9}\), which is the simplest form of the original radical expression.
By applying the square root to both the top and bottom numbers individually, they can each be simplified if possible, and then the results form a new, often simpler, fraction. In our exercise, applying the rule to \(\frac{1}{81}\), we evaluate \(\sqrt{1}\) and \(\sqrt{81}\) separately to obtain \(\frac{1}{9}\), which is the simplest form of the original radical expression.
Radical Expressions
Radical expressions are expressions that contain a root symbol, and the most common type is the square root. The goal when simplifying radical expressions is to find the most simplified form of the expression, ensuring that no perfect squares remain under the radical sign if possible. Simplifying radicals can involve several steps including identifying and factoring out perfect squares, fractionating the radicands (the numbers under the root sign), and rationalizing the denominator if necessary.
In dealing with radical expressions, remember to consider the properties of radicals that allow for the manipulation and simplification of these expressions, like the quotient rule applied in our exercise. The quotient rule makes handling radical expressions involving fractions much more manageable, thus highlighting the importance of understanding how to negotiate square roots, whether they are part of whole numbers or fractions.
In dealing with radical expressions, remember to consider the properties of radicals that allow for the manipulation and simplification of these expressions, like the quotient rule applied in our exercise. The quotient rule makes handling radical expressions involving fractions much more manageable, thus highlighting the importance of understanding how to negotiate square roots, whether they are part of whole numbers or fractions.
Other exercises in this chapter
Problem 16
rewrite each expression without absolute value bars. $$ |-203| $$
View solution Problem 16
Find each product. $$(x+5)\left(x^{2}-5 x+25\right)$$
View solution Problem 17
Evaluate each exponential expression. $$ \frac{2^{8}}{2^{4}} $$
View solution Problem 17
rewrite each expression without absolute value bars. $$ |12-\pi| $$
View solution