Problem 50
Question
For the following exercises, simplify each expression. \(\sqrt{\frac{20}{121 a^{4}}}\)
Step-by-Step Solution
Verified Answer
The simplified expression is \(\frac{2\sqrt{5}}{11a^2}\).
1Step 1: Split the Radicand
Start by breaking down the radicand inside the square root into simpler components. Split the fraction \[\frac{20}{121a^4}\]into \[\frac{20}{121} \times \frac{1}{a^4}.\]
2Step 2: Simplify the Fraction with Square Roots
Calculate the square root of each component individually. First, consider the fraction \[\frac{20}{121}.\] You can write it as \[\frac{\sqrt{20}}{\sqrt{121}}.\] Then simplify \[\sqrt{121}\] to \[11\] because \(11^2 = 121\).
3Step 3: Simplify the Radicals
Simplify \(\sqrt{20}\) further. Notice that \(20 = 4 \times 5\), which can be split as:\[\sqrt{4} \times \sqrt{5}.\] Then, \(\sqrt{4}\) simplifies to \(2\), so \(\sqrt{20} = 2\sqrt{5}\).
4Step 4: Combine Results
Now combine the simplified parts back into the expression. The expression becomes:\[\frac{2\sqrt{5}}{11}.\] This represents the simplified form of the radicals in terms of the numerator.
5Step 5: Simplify the Algebraic Component
The remaining part is \(\sqrt{\frac{1}{a^4}}\), which can be rewritten as:\[\frac{1}{\sqrt{a^4}}.\] The square root of \(a^4\) is \(a^2\), hence:\[\frac{1}{a^2}.\]
6Step 6: Combine All Parts
Combine the simplified numerical and algebraic expressions. The complete simplified expression is \[\frac{2\sqrt{5}}{11a^2}.\]
Key Concepts
Square RootsRadicalsAlgebraic FractionsExponents
Square Roots
A square root asks what number, when multiplied by itself, will give the original number. When simplifying expressions with square roots, like in this exercise, we often break down the number inside the square root into its factors.
For example, the square root of 20 can be split into two easier parts: 4 and 5, since 4 times 5 is 20. The number 4 is a perfect square because its square root is an integer: 2, because 2 times 2 equals 4.
Therefore, \[ \sqrt{20} = \sqrt{4} imes \sqrt{5} = 2\sqrt{5}. \] This step allows us to simplify the problem into simpler pieces.
For example, the square root of 20 can be split into two easier parts: 4 and 5, since 4 times 5 is 20. The number 4 is a perfect square because its square root is an integer: 2, because 2 times 2 equals 4.
Therefore, \[ \sqrt{20} = \sqrt{4} imes \sqrt{5} = 2\sqrt{5}. \] This step allows us to simplify the problem into simpler pieces.
Radicals
Radicals encompass more than just square roots; they're any expression containing a root. In this problem, the radical is the square root of a fraction. Breaking it down involves both the numerator and the denominator.
First, we separate each element within the radical. Taking the fraction \( \frac{20}{121} \), we can rewrite it using square roots as \( \frac{\sqrt{20}}{\sqrt{121}} \). Each part of the fraction can then be simplified individually.
Simplifying radicals is about identifying perfect squares. For instance, 121 is a perfect square because 11 times 11 equals 121. So, \(\sqrt{121} = 11\). By using the property that a radical of a quotient is the quotient of the radicals, we simplify our work on each separate part.
First, we separate each element within the radical. Taking the fraction \( \frac{20}{121} \), we can rewrite it using square roots as \( \frac{\sqrt{20}}{\sqrt{121}} \). Each part of the fraction can then be simplified individually.
Simplifying radicals is about identifying perfect squares. For instance, 121 is a perfect square because 11 times 11 equals 121. So, \(\sqrt{121} = 11\). By using the property that a radical of a quotient is the quotient of the radicals, we simplify our work on each separate part.
Algebraic Fractions
An algebraic fraction is a fraction with variables in its numerator, denominator, or both. In this problem, we handle an algebraic term \( \frac{1}{a^4} \).
When a variable is under a square root, we employ exponent rules to simplify it. Square roots are the same as raising to the power of \( \frac{1}{2} \). Hence, the denominator \( a^4 \) under a square root becomes \( a^4 \) raised to the power of \( \frac{1}{2} \), which equals \( a^2 \).
Therefore, \( \frac{1}{\sqrt{a^4}} = \frac{1}{a^2} \). This simplification is key when dealing with complex algebraic fractions in radical form.
When a variable is under a square root, we employ exponent rules to simplify it. Square roots are the same as raising to the power of \( \frac{1}{2} \). Hence, the denominator \( a^4 \) under a square root becomes \( a^4 \) raised to the power of \( \frac{1}{2} \), which equals \( a^2 \).
Therefore, \( \frac{1}{\sqrt{a^4}} = \frac{1}{a^2} \). This simplification is key when dealing with complex algebraic fractions in radical form.
Exponents
Exponents are shorthand for repeated multiplication. They are crucial when simplifying expressions involving roots. For instance, \( a^4 \) means multiplying \( a \) by itself four times.
When dealing with exponents and square roots together, remember that a square root is equivalent to raising to the \( \frac{1}{2} \) power. So, \( \sqrt{a^4} \) becomes \( (a^4)^{\frac{1}{2}} \). Simplifying yields \( a^{4 \times \frac{1}{2}} = a^2 \).
This compliance with exponent rules allows the complete simplification process of complex algebraic expressions. Understanding exponents helps manage terms efficiently, reducing them to their simplest forms.
When dealing with exponents and square roots together, remember that a square root is equivalent to raising to the \( \frac{1}{2} \) power. So, \( \sqrt{a^4} \) becomes \( (a^4)^{\frac{1}{2}} \). Simplifying yields \( a^{4 \times \frac{1}{2}} = a^2 \).
This compliance with exponent rules allows the complete simplification process of complex algebraic expressions. Understanding exponents helps manage terms efficiently, reducing them to their simplest forms.
Other exercises in this chapter
Problem 50
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