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Chapter 7: Problem 63

Rationalize each denominator. Assume that all variables represent positive real numbers. $$ \sqrt{\frac{288 x^{7}}{y^{9}}} $$

Short Answer

Expert verified
\(\frac{12 x^{7/2} \sqrt{2}}{y^9}\)

Step by step solution

01

Simplify the expression under the square root

Rewrite the given expression \(\frac{288 x^{7}}{y^{9}}\) such that it separates into parts that can be simplified separately. Each part should be addressed as \(\sqrt{288} \frac{x^7}{y^9}\).
02

Simplify \(\sqrt{288}\)

\(288 = 2^5 \times 3^2\). Therefore, \(\sqrt{288} = \sqrt{2^5 \times 3^2} = \sqrt{(2^2)^2 \times 2 \times 3^2} = 2^2 \times 3 \times \sqrt{2} = 12 \sqrt{2}\)
03

Simplify the variables

Combine the simplified \( \sqrt{288} = 12 \sqrt{2} \) with \( x^7 \) and \( y^9 \). Rewrite \( \frac{x^7}{y^9} \) as \( x^{7/2} \times y^{-9/2} \) under the square root.
04

Rationalize the denominator

To eliminate the square root in the denominator, multiply both numerator and denominator by \( y^{9/2} \) to get \( \frac{12 \sqrt{2} \times x^{7/2} \times y^{9/2}}{y^{9/2} \times y^{9/2}} \). This results in \( 12 \cdot x^{7/2} \cdot y^{9/2} \sqrt{2} = 12 y^{9/2} \).
05

Combine all simplified parts

Combine the parts to get the final simplified and rationalized expression: \(\frac{12 x^{7/2} y^{9/2} \sqrt{2}}{y^9}\). Simplify further to find the rationalized form.

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Key Concepts

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

Simplifying Expressions
Simplifying expressions means rewriting them in a simpler or more efficient form. For example, in the exercise given, we simplified the expression under the square root.
  • Understand the components: Break down complex expressions into simpler parts.
  • Factorization: Express numbers and variables in a factored form to easily simplify later.
Consider \(\frac{288 x^{7}}{y^{9}}\). By breaking this down, we can view it as \(\frac{\textbf{Separate parts}}{\textbf{for simplification}}\). Simplification makes complex algebra readable and easy to manipulate.
Square Roots
Square roots are a fundamental operation in algebra. Taking the square root of a number \(x\) means finding a number that, when multiplied by itself, gives \(x\).
  • To simplify \( \sqrt{288} \), break down the number: \( 288 = 2^5 \times 3^2 \).
  • Use properties: \( \sqrt{288} = \sqrt{(2^2)^2 \times 2 \times 3^2} = 2^2 \times 3 \times \sqrt{2} = 12 \sqrt{2} \).
This way, \( \sqrt{288} \) simplifies to a form combined with the square root of a smaller number and a multiplication of perfect squares.
Algebraic Fractions
Algebraic fractions involve variables in the numerator, the denominator, or both. Simplifying them often requires rationalizing the denominator, finding common denominators, and reducing factors.
  • Rewrite complex fractions: \( \frac{x^7}{y^9} = x^{7/2} \times y^{-9/2} \), makes future steps clearer.
  • Rationalize: To eliminate radicals from the denominator, multiply both numerator and denominator by an appropriate factor, here it's \( y^{9/2} \).
In our exercise, we multiply by \( y^{9/2} \) to rationalize, making the expression free of square roots in the denominator.
Exponents
Exponents describe how many times a number, or variable, is multiplied by itself. They follow specific properties that help simplify complex expressions.
  • Product rule: \( a^m \cdot a^n = a^{m+n} \).
  • Power rule: \( (a^m)^n = a^{mn} \).
In the context of our exercise: \( \frac{x^7}{y^9} = x^{7/2} \times y^{-9/2} \). This expression shows how exponents are used to rewrite the fraction and how they are used in simplifying further steps.

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Most popular questions from this chapter

Rationalize each denominator. Assume that all variables represent positive real numbers. $$ -\sqrt{\frac{75 m^{3}}{p}} $$

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