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Chapter 9: Problem 53

Simplify the radical expression. $$\sqrt{\frac{x^{6}}{16 y^{2}}}$$

Short Answer

Expert verified
The simplified form of the given radical expression \(\sqrt{\frac{x^{6}}{16 y^{2}}}\) is \(\frac{x^{3}}{4y}\)

Step by step solution

01

Breaking Down the Radical Expression

Apply the rule of square roots to break the expression down. This results in: \(\sqrt{x^{6}}/\sqrt{16y^{2}}\)
02

Simplifying the Square Root of the Numerator

The square root of \(x^{6}\) is \(x^{3}\) because \(x^{2}\) raised to the third power equals \(x^{6}\). Therefore, \(\sqrt{x^{6}}\) simplifies to \(x^{3}\).
03

Simplifying the Square Root of the Denominator

The square root of \(16y^{2}\) is \(4y\) because \(4y\) squared equals \(16y^{2}\). So, \(\sqrt{16y^{2}}\) simplifies to \(4y\).
04

Combining the Simplified Expressions

Combine the two simplified expressions. This results in: \(x^{3}/4y\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Roots
In mathematics, a square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 times 4 equals 16. The concept of square roots is fundamental when working with radical expressions.

To simplify
  • Start by breaking down the expression inside the radical
  • Apply the property of square roots that states: \(\sqrt{a\cdot b} = \sqrt{a}\cdot\sqrt{b}\)
In the given exercise, \(\sqrt{\frac{x^6}{16y^2}}\), you apply this property to separate the numerator and the denominator (more on those later) inside the square root, leading us to \(\sqrt{x^6}/\sqrt{16y^2}\). This separation is key to simplifying the expression further.
Numerator and Denominator
In any fraction, such as \(\frac{x^3}{4y}\) from our original problem, the number on top is called the numerator, and the number on the bottom is called the denominator. Understanding these terms is crucial because they form the foundation for operations involving fractions or rational expressions.

When you simplify fractions, you perform operations separately on the numerator and the denominator.
  • For \(\sqrt{x^6}\), the square root simplifies the numerator to \(x^3\). This is because \((x^3)^2 = x^6\).
  • For \(\sqrt{16y^2}\), the square root simplifies the denominator to \(4y\). This is calculated as \((4y)^2 = 16y^2\).
Finally, connect them back together to form the newly simplified expression \(\frac{x^3}{4y}\). Each step of this simplification process relies on understanding how to independently handle the numerator and the denominator.
Algebraic Expressions
Algebraic expressions are composed of variables, numbers, and operations. They are fundamental components in algebra and help us represent real-world problems through mathematical symbols and operations. In the problem, expressions like \(x^6\) and \(16y^2\) are part of a larger radical expression.

To work effectively with these expressions:
  • Identify the variables and their exponents in the expression. For example, \(x^6\) suggests the variable \(x\) is raised to the sixth power.
  • Look for opportunities to simplify by factoring or breaking down expressions to their base components, like transforming \(x^6\) into \((x^3)^2\).
  • Apply algebraic rules and properties, such as the distributive property, to manipulate and simplify expressions as needed.
Through this process, you can simplify complex expressions, making it easier to solve more significant algebraic problems.

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Most popular questions from this chapter

Rationalize the denominator of the expression and simplify. (Assume all variables are positive.) $$\frac{9}{\sqrt[3]{4}}$$

Find the distance between the two points. Round your answer to two decimal places, if necessary. $$(15,7),(3,5)$$

Determine whether the lines are parallel, perpendicular, or neither. $$ \begin{aligned} &y=\frac{1}{2} x+7 \\ &y=2 x+7 \end{aligned} $$

Rewrite the expression as a single fraction and simplify. $$3-\frac{1}{\sqrt{3}}$$

Logic Describe the steps used to solve a radical equation.
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