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Chapter 7: Problem 52

Rationalize each denominator. Assume that all variables represent positive numbers. $$ \frac{\sqrt[5]{3 a^{4}}}{\sqrt[5]{2 b^{7}}} $$

Short Answer

Expert verified
\( \frac{\root{5}{48 a^4 b^{28}}}{2 b^7} \)

Step by step solution

01

Identify the Problem

The given expression is \(\frac{\root{5}{3 a^{4}}}{\root{5}{2 b^{7}}}\). The problem requires rationalizing the denominator.
02

Determine What to Multiply By

To rationalize the denominator, multiply both the numerator and the denominator by \(\root{5}{(2 b^7)^4}\).
03

Multiply Numerator and Denominator

Multiplying both the numerator and the denominator: \(\frac{\root{5}{3 a^4} \times \root{5}{(2 b^7)^4}}{\root{5}{2 b^7} \times \root{5}{(2 b^7)^4}} = \frac{\root{5}{3 a^4 \times (2 b^7)^4}}{\root{5}{(2 b^7)^5}} \).
04

Simplify the Denominator

Simplify the denominator: \(\root{5}{(2 b^7)^5} = 2 b^7\).
05

Simplify the Numerator

Simplify the numerator: \(\root{5}{3 a^4 \times (2 b^7)^4} = \root{5}{3 a^4 \times 16 b^{28}} = \root{5}{48 a^4 b^{28}}\).
06

Final Expression

Putting it all together, the final expression is \(\frac{\root{5}{48 a^4 b^{28}}}{2 b^7}\).

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Key Concepts

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

radicals
A radical is a symbol that denotes the root of a number. For example, the square root of a number is represented as \(\root{2}{x}\) or simply \( \sqrt{x} \). This can be extended to cube roots \( \root{3}{x} \) or in this case, fifth roots, \( \root{5}{x} \).
When working with radicals, it鈥檚 important to understand how they operate:
  • Radicals can be simplified by factoring out perfect roots.
  • You can multiply radicals with the same index (e.g., \( \root{n}{a} \times \root{n}{b} = \root{n}{ab} \)).
  • Rationalizing the denominator involves getting rid of radicals in the denominator.
Radicals are commonly used in algebra to solve roots and to simplify complex expressions.
rationalizing the denominator
Rationalizing the denominator means removing any radical expressions from the denominator of a fraction. Here are the steps involved:
  • Identify the Radical: First, recognize the radical in the denominator. In our example, it's \( \root{5}{2 b^7} \).
  • Determine What to Multiply By: Multiply the numerator and the denominator by a term that will eliminate the radical. For example, we multiply by \( \root{5}{ (2 b^7)^4 } \) to make the denominator a perfect fifth power.
  • Perform the Multiplication: Both numerator and denominator are multiplied. This translates to multiplying radicals properly.
  • Simplify: The denominator simplifies into a rational number because the powers now match the radical's index (fifth root in this case).
Rationalizing the denominator is essential because it helps in simplifying the expression and making it easier to work with.
exponents
Exponents represent how many times a number (the base) is multiplied by itself. For instance, in \( a^4 \), the base is \a\ and the exponent is \4\鈥攎eaning \a\ is multiplied by itself 4 times.
When dealing with expressions involving radicals and exponents, it鈥檚 crucial to understand their properties:
  • Multiplication of exponents: \( a^m \times a^n = a^{m+n} \).
  • Power of a power: \( (a^m)^n = a^{mn} \).
  • Division of exponents: \( a^m / a^n = a^{m-n} \).
  • When the same base is raised to a fractional exponent: \( a^{m/n} = \root{n}{a^m} \)
In the exercise, we used these exponent rules to simplify both the numerator and the denominator. Understanding how to manipulate exponents is critical in rationalizing the denominator and in simplifying radicals.

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Most popular questions from this chapter

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