Problem 71
Question
Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt{\frac{x^{20}}{4 y^{2}}} $$
Step-by-Step Solution
Verified Answer
\(\frac{x^{10}}{2y}\)
1Step 1: Simplify the Constant
First, we simplify the radical by separating the constant from the variables: \[\sqrt{\frac{x^{20}}{4y^2}} = \frac{1}{2} \sqrt{\frac{x^{20}}{y^2}}. \]This is because \(\sqrt{4} = 2\).
2Step 2: Simplify the Variables Inside the Square Root
The expression inside the square root is \(\frac{x^{20}}{y^2}\). Since \(\sqrt{x^{20}} = x^{10}\) and \(\sqrt{y^2} = y\), it can be simplified to:\[\frac{x^{10}}{y}.\]
3Step 3: Combine the Results
Combine the simplifications from Steps 1 and 2:\[\frac{1}{2} \cdot \frac{x^{10}}{y} = \frac{x^{10}}{2y}.\]This is the simplified expression.
Key Concepts
Understanding Square RootsVariables in AlgebraExponents in Algebra
Understanding Square Roots
A square root is a number that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2, because 2 times 2 is 4. The square root is represented by the symbol \( \sqrt{} \). Understanding square roots is essential because they allow us to simplify expressions in algebra, especially when dealing with powers and radicals like the one in the exercise.
In the exercise, we had \( \sqrt{4} \) as part of the expression. The square root of 4 is straightforward, which is 2. Simplifying this helps to break down the expression into more manageable parts, making it easier to work with variables and exponents later on.
In the exercise, we had \( \sqrt{4} \) as part of the expression. The square root of 4 is straightforward, which is 2. Simplifying this helps to break down the expression into more manageable parts, making it easier to work with variables and exponents later on.
Variables in Algebra
Variables in algebra are symbols, usually letters, that represent unknown or changing numbers. For example, in the expression \( x^{20} \), \( x \) is a variable. Variables can stand for any number, but they make algebra very flexible when solving problems.
In the context of the exercise, both \( x \) and \( y \) are variables representing positive real numbers. When simplifying the radical \( \sqrt{\frac{x^{20}}{4y^2}} \), it's important to treat these variables correctly. The step-by-step solution simplifies \( x^{20} \) to \( x^{10} \), based on the rule that \( \sqrt{x^n} = x^{n/2} \). Understanding how to handle variables within square roots allows you to manipulate and simplify complex expressions effectively.
In the context of the exercise, both \( x \) and \( y \) are variables representing positive real numbers. When simplifying the radical \( \sqrt{\frac{x^{20}}{4y^2}} \), it's important to treat these variables correctly. The step-by-step solution simplifies \( x^{20} \) to \( x^{10} \), based on the rule that \( \sqrt{x^n} = x^{n/2} \). Understanding how to handle variables within square roots allows you to manipulate and simplify complex expressions effectively.
Exponents in Algebra
Exponents are a key part of algebra. They describe how many times a number, the base, is multiplied by itself. For instance, \( x^{20} \) means \( x \) is multiplied by itself 20 times. Exponents simplify the expression of large numbers and enable us to perform quick calculations.
To simplify expressions with exponents under square roots, we use the property that \( \sqrt{x^n} = x^{n/2} \). This is exactly what was done in the exercise to simplify \( \sqrt{x^{20}} \) to \( x^{10} \).
To simplify expressions with exponents under square roots, we use the property that \( \sqrt{x^n} = x^{n/2} \). This is exactly what was done in the exercise to simplify \( \sqrt{x^{20}} \) to \( x^{10} \).
- This property helps reduce the complexity of algebraic expressions.
- By dividing the exponent by 2, you effectively simplify the expression, making it easier to work with.
Other exercises in this chapter
Problem 70
Perform each indicated operation. Write the result in the form \(a+b i\). $$ \frac{5}{3-2 i} $$
View solution Problem 70
Multiply and then simplify if possible. $$ (\sqrt[3]{3}+\sqrt[3]{2})(\sqrt[3]{9}-\sqrt[3]{4}) $$
View solution Problem 71
Factor the given factor from the expression. $$ x^{-1 / 3} ; 5 x^{-1 / 3}+x^{2 / 3} $$
View solution Problem 71
Perform each indicated operation. Write the result in the form \(a+b i\). $$ (5-6 i)-4 i $$
View solution