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Chapter 10: Problem 51

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\frac{\sqrt[3]{5 y^{2}}}{\sqrt[3]{4 x}}\)

Short Answer

Expert verified
The rationalized expression is \(\frac{5y^2}{\sqrt[3]{100xy^4}}\).

Step by step solution

01

Identify the Need to Rationalize

To rationalize the numerator of the expression \(\frac{\sqrt[3]{5 y^{2}}}{\sqrt[3]{4 x}}\), we need to eliminate the cube root in the numerator. This is done by multiplying the numerator and the denominator by a term that makes the expression a perfect cube.
02

Determine the Rationalizing Factor

Since the numerator is \(\sqrt[3]{5 y^2}\), to make it a perfect cube, we need another factor of \(5y^2\). Therefore, the rationalizing factor is \(5^{2/3} y^{4/3}\).
03

Multiply to Eliminate the Cube Root

Multiply both the numerator and the denominator by \(\sqrt[3]{25 y^4}\) (which is the decimal representation of \((5y^2)^{2/3}\)): \[\frac{\sqrt[3]{5 y^2} \cdot \sqrt[3]{25 y^4}}{\sqrt[3]{4 x} \cdot \sqrt[3]{25 y^4}}\] This simplifies to \[\frac{\sqrt[3]{(5 y^2)^3}}{\sqrt[3]{100 x y^4}}\]
04

Simplify the Expression

The cube root of the numerator, where the expression is \((5y^2)^3\), becomes \(5y^2\). Therefore, the numerator simplifies to 5y^2. The denominator becomes \(\sqrt[3]{100x y^4}\).

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Key Concepts

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

Rationalization Process
Rationalization is a technique used in mathematics to simplify expressions involving roots. In our problem, this involves removing the cube root from the numerator.
  • This is done by multiplying both the numerator and denominator by a strategic value, called the rationalizing factor.
  • By making the numerator a perfect cube, we eliminate the root without altering the overall value of the fraction.
This is crucial in expressions, as it often results in a cleaner, more manageable form. Imagine rationalization as bringing order to chaos鈥攁 handy trick for transforming complex fractional expressions into simpler ones.
Cube Roots
A cube root asks what number, multiplied by itself three times, gives us the original value.
  • For instance, the cube root of 8 is 2, because 2 multiplied by itself thrice is 8.
  • In cube root expressions, we often need to adjust them to reach a simpler form.
In our task, the numerator is \(\sqrt[3]{5y^2}\), which represents a less tidy form that we wish to clean up through rationalization. By multiplying the expression in such a way that our numerator becomes a perfect cube, the root vanishes, transforming it into a more straightforward numerical value.
Mathematical Expressions
Mathematical expressions like the one given in the problem can at first seem a bit daunting, but they are simply representations of numbers and operations performed on them.
  • In our particular case, \(\frac{\sqrt[3]{5y^2}}{\sqrt[3]{4x}}\), we have a division of two cube root expressions.
  • Understanding these expressions requires knowing the interplay of numbers, roots, and operations.
The goal is to manipulate these expressions in a way that simplifies them while maintaining their equality. By focusing on the roots and applying orderly mathematical operations, we simplify complex problems efficiently.
Multiplying Rational Expressions
Multiplying rational expressions is a key part of the rationalization process.
  • This involves combining fractions by multiplying both numerators and denominators.
  • Let's consider our example: we multiply the expression by \(\sqrt[3]{25y^4}\) on both top and bottom.
The trick is to choose a multiplying factor that simplifies the expression.By doing this, we exploit the properties of exponents and roots to cancel the cube root in the numerator, resulting in a rational expression. Understanding how to choose these multipliers and what they achieve is integral to mastering multiplying rational expressions.

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Most popular questions from this chapter

The maximum distance \(D(h)\) in kilometers that a person can see from a height h kilometers above the ground is given by the function \(D(h)=111.7 \sqrt{h}\). Use this function for Exercises 79 and 80. Round your answers to two decimal places. Find the height that would allow a person to see 80 kilometers.

Simplify. $$ \frac{\frac{z}{5}+\frac{1}{10}}{\frac{z}{20}-\frac{z}{5}} $$

Solve. $$ \sqrt{3 x+1}-2=0 $$

Solve. $$ \sqrt[4]{4 x+1}-2=0 $$

Solve. $$ \sqrt{3 x+4}=5 $$
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