/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 458 In the following exercises, simp... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 458

In the following exercises, simplify. (a) \(\sqrt[4]{16 x^{8}}\) (b) \(\sqrt[6]{64 y^{12}}\)

Short Answer

Expert verified
a) 2x虏 b) 2y虏

Step by step solution

01

Simplify the Expression (a) \( \sqrt[4]{16 x^{8}} \)

First, factor 16 as a power of 4: \(16 = 2^4\). Therefore, \( \sqrt[4]{16} = \sqrt[4]{2^4} = 2\). For the variable part, use the property of radicals: \( \sqrt[4]{x^8} = (x^8)^{1/4} = x^{8/4} = x^2 \). Thus, the simplified form is \(2 x^2 \).
02

Simplify the Expression (b) \( \sqrt[6]{64 y^{12}} \)

First, factor 64 as a power of 6: \(64 = 2^6\). Therefore, \( \sqrt[6]{64} = \sqrt[6]{2^6} = 2\). For the variable part, use the property of radicals: \( \sqrt[6]{y^{12}} = (y^{12})^{1/6} = y^{12/6} = y^2 \). Thus, the simplified form is \(2 y^2 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Understanding Radicals
Radicals are the opposite of exponents. When you see a radical, it asks for which number, when raised to a certain power, gives the number inside the radical. For example, \(\sqrt{9}\) asks which number multiplied by itself gives 9, and the answer is 3. Radicals can also involve higher roots, like cube roots (\(\sqrt[3]{...}\)) and fourth roots (\(\sqrt[4]{...}\)).

In the exercise, simplifying \(\sqrt[4]{16 x^{8}}\) and \(\sqrt[6]{64 y^{12}}\) involves higher roots. Since radicals and exponents share properties, we can use the relationship between them for simplification.
Working with Exponents
Exponents tell you how many times a number should be multiplied by itself. For example, \(3^2 = 3 \times 3 = 9\). There are also fractional exponents, where the denominator signifies the root. So, \(x^{1/n}\) is the same as \(\sqrt[n]{x}\).

In our exercise, it's crucial to convert the radicals into their exponent form. For \(\sqrt[4]{16 x^{8}}\), we express 16 as \(2^4\) and the fourth root becomes \(2^{4/4} = 2\). For the variable part, \(x^8\) under the fourth root becomes \(x^{8/4} = x^2\). The same logic applies to \(64 y^{12}\), where converting 64 to \(2^6\) and taking the sixth root gives \(2\). For the variable part, \(y^{12}\) under the sixth root simplifies to \(y^2\).
Algebraic Simplification
Algebraic simplification is about making expressions as simple as possible. Simplifying radicals often involves expressing numbers as powers and reducing them using exponent rules.

Let's break down the steps:
  • Identify the base number and its exponent
  • Express the base number as a power if not already done
  • Apply the root by converting it to a fractional exponent
  • Simplify the exponents.
In the first example, \(\sqrt[4]{16} \equiv 2\) and \(\sqrt[4]{x^8} \equiv x^2\), so combining them gives \(2x^2\).
In the second example, \(\sqrt[6]{64} \equiv 2\) and \(\sqrt[6]{y^{12}} \equiv y^2\). Combining them results in \(2y^2\). Thus, simplifying radicals involves careful application of exponent rules, making algebraic expressions easier to handle.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In the following exercises, simplify. $$ \frac{\sqrt{15 x^{3}}}{\sqrt{3 x}} $$

In the following exercises, simplify and rationalize the denominator. $$ \sqrt{\frac{3}{20}} $$

In the following exercises, solve. (a) \(\sqrt{s-8}-s=-8\) (b) \(\sqrt{w+25}-w+5=0\)

In the following exercises, simplify. $$ \frac{\sqrt{26 y^{7}}}{\sqrt{2 y}} $$

In the following exercises, solve. \(\sqrt{2 y+4}+6=0\)
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.