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Chapter 1: Problem 13

Simplify the expression, and rationalize the denominator when appropriate. $$\frac{\sqrt{12 x^{4} y}}{\sqrt{3 x^{2} y^{5}}}$$

Short Answer

Expert verified
The simplified expression is \( \frac{2x}{y^2} \).

Step by step solution

01

Rewrite the Radicals

Start by breaking down the radicals in the numerator and denominator. We have:\[ \frac{\sqrt{12 x^4 y}}{\sqrt{3 x^2 y^5}} = \frac{\sqrt{12} \cdot \sqrt{x^4} \cdot \sqrt{y}}{\sqrt{3} \cdot \sqrt{x^2} \cdot \sqrt{y^5}}. \]
02

Simplify Each Component

Simplify each square root separately. First, simplify \(\sqrt{12}\) and \(\sqrt{3}\):\[ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \] and \[ \sqrt{3} = \sqrt{3}. \] Now simplify \(\sqrt{x^4}\) and \(\sqrt{x^2}\):\[ \sqrt{x^4} = x^2 \] and \[ \sqrt{x^2} = x. \] For \(\sqrt{y}\) and \(\sqrt{y^5}\), write them as powers of \(y\):\[ \sqrt{y} = y^{1/2} \] and \[ \sqrt{y^5} = y^{5/2}. \]
03

Rewrite the Expression

Substitute the simplified components back into the expression:\[ \frac{2\sqrt{3} \cdot x^2 \cdot y^{1/2}}{\sqrt{3} \cdot x \cdot y^{5/2}}. \]
04

Cancel Common Factors

Cancel out any common factors from the numerator and the denominator. The common factor here is \(\sqrt{3}\), leaving us with:\[ \frac{2x^2y^{1/2}}{xy^{5/2}}. \] Simplify further by canceling \(x\) and using the property of exponents:\[ 2x^{2-1}y^{1/2-5/2} = 2x^{1}y^{-2} = 2x\cdot \frac{1}{y^2}. \]
05

Simplify Further if Possible

Ensure the expression is simplified further so no negative exponents remain: \[ \frac{2x}{y^2}. \] This is the expression with a rationalized denominator.

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Key Concepts

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

Simplification of Expressions
In mathematics, simplifying expressions is a crucial skill. It involves rewriting an expression in its simplest form. This process usually makes the expression more comprehensible and easier to work with.
For the exercise at hand, we began with a fraction involving square roots, which needed simplification.
  • Separate Elements: First, break down complex radicals into simpler parts. This might involve identifying perfect squares within a square root.
  • Combine Like Terms: Once simplified, the component parts can be recombined, keeping in mind to reduce where possible. Cancellation occurs if both the numerator and denominator share similar terms.
The primary goal here is to present the expression in its simplest equivalent form, so it's more manageable.
Properties of Exponents
The properties of exponents are like a toolbox for manipulating expressions involving powers. They help us simplify and evaluate radicals effectively.
  • Product Rule: This property states that when you multiply two powers with the same base, you add their exponents: \(a^m \cdot a^n = a^{m+n}\).
  • Quotient Rule: When dividing powers with the same base, you subtract the exponents: \(a^m / a^n = a^{m-n}\).
  • Negative Exponent Rule: It indicates a reciprocal relationship: \(a^{-n} = \frac{1}{a^n}\).
Applying these rules, we simplified \(x^{2-1}\) to \(x\) and transformed \(y^{1/2 - 5/2}\) to \(y^{-2}\). Rewriting negative exponents as positive ones is important for further simplification.
Radical Expressions
Radical expressions involve roots, such as square roots or cube roots. Simplifying radical expressions often means rationalizing them, removing any roots from the denominator.
  • Breaking Down Radicals: We started by extracting square roots of numbers, like \(\sqrt{12}\) being expressed as \(2\sqrt{3}\).
  • Rationalizing the Denominator: This process removes radicals from a fraction's denominator. Multiply both the numerator and denominator by the necessary radical value to achieve this.
In our exercise, rationalizing was while ensuring no radicals remained in the denominator. Squeezing through these steps demands precision and clarity to make the expression easier to work with.

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Most popular questions from this chapter

A person's body surface area \(S\) (in square feet) can be approximated by $$S=(0.1091) w^{1423} h^{0.723}$$ where height \(h\) is in inches and weight \(w\) is in pounds. (a) Estimate \(S\) for a person 6 feet tall weighing 175 pounds. (b) If a person is 5 feet 6 inches tall, what effect does a \(10 \%\) increase in weight have on \(S ?\)

A large grain silo is to be constructed in the shape of a circular cylinder with a hemisphere attached to the top (see the figure). The diameter of the silo is to be 30 feet, but the height is yet to be determined. Find the height \(h\) of the silo that will result in a capacity of \(11,250 \pi \mathrm{ft}^{3}\).

Replace the symbol \(\square\) with either \(=\) or \(\neq\) to make the resulting statement true, whenever the expression has meaning. Give a reason for your answer. $$a^{x} b^{y} \square(a b)^{x y}$$

A bullet is fired horizontally at a target, and the sound of its impact is heard 1.5 seconds later. If the speed of the bullet is \(3300 \mathrm{ft} / \mathrm{sec}\) and the speed of sound is \(1100 \mathrm{ft} / \mathrm{sec},\) how far away is the target?

Archeologists can determine the height of a human without having a complete skeleton. If an archeologist finds only a humerus, then the height of the individual can be determined by using a simple linear relationship. (The humerus is the bone between the shoulder and the elbow.) For a female, if \(x\) is the length of the humerus (in centimeters), then her height \(h\) (in centimeters) can be determined using the formula \(h=65+3.14 x .\) For a male, \(h=73.6+3.0 x\) should be used. (a) A female skeleton having a 30 -centimeter humerus is found. Find the woman's height at death. (b) A person's height will typically decrease by 0.06 centimeter each year after age \(30 .\) A complete male skeleton is found. The humerus is 34 centimeters, and the man's height was 174 centimeters. Determine his approximate age at death.
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