/*! 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 455 In the following exercises, simp... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 455

In the following exercises, simplify. (a) \(\sqrt[5]{a^{10}}\) (b) \(\sqrt[3]{b^{27}}\)

Short Answer

Expert verified
(a) is \(\sqrt[5]{a^{10}}\) simplified to \({a^2}\), and (b) is \(\sqrt[3]{b^{27}}\) simplified to \({b^9}\).

Step by step solution

01

Understanding the Problem

The task is to simplify two expressions involving roots and exponents. The given expressions are: (a) \(\sqrt[5]{a^{10}}\) (b) \(\sqrt[3]{b^{27}}\)
02

Simplify \(\sqrt[5]{a^{10}}\)

The n-th root of a power can be expressed in terms of fractional exponents. Specifically, \(\sqrt[n]{x^m} = x^{m/n}\). So, applying this rule to \(\sqrt[5]{a^{10}}\): \(a^{10/5} = a^2\).
03

Simplify \(\sqrt[3]{b^{27}}\)

Similarly, for the cube root, the rule \(\sqrt[n]{x^m}\) is applied as follows: \(\sqrt[3]{b^{27}} = b^{27/3} = b^9\).

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.

Fractional Exponents
When you see an expression involving roots, you can often simplify it using fractional exponents. Fractional exponents are another way of expressing roots. An expression like \(\frac{m}{n}\) in the exponent is equivalent to the n-th root of the base raised to the m-th power. For example, \[ x^{m/n} = \sqrt[n]{x^m} \]
This means you can convert roots into a more manageable form using exponents.
nth Roots
The concept of nth roots is central to simplifying expressions like \(\sqrt[n]{x^m}\). The nth root of a number is the number which, when raised to the power of n, gives the original number. For instance, the fifth root of a number y (written as \(\sqrt[5]{y}\)) is the number which, when multiplied by itself five times, equals y.
To express this in terms of exponents, you write it as \[ y^{1/5} \]. This transformation makes it easier to work with large exponents, as you can apply exponent rules more straightforwardly.
Simplifying Expressions
Simplifying expressions involving roots and exponents involves converting the radical form to fractional exponents and then reducing those exponents. The rule \(\sqrt[n]{x^m} = x^{m/n}\) is very helpful.
For example, if you have the expression \(\sqrt[5]{a^{10}}\), you convert it first to fractional exponents: \[ a^{10/5} \]. Since \[ 10/5 = 2\], the expression simplifies to \[ a^2 \]. This makes the problem straightforward and easier to handle.
Cube Roots
Cube roots are a specific case of nth roots where n=3. The cube root of a number b (written as \(\sqrt[3]{b}\)) is a number which, when multiplied by itself three times, gives b. In terms of exponents, this is written as \[ b^{1/3} \].
This notation helps in reducing larger exponents efficiently. For example, with \(\sqrt[3]{b^{27}}\), converting it to fractional exponents gives \[ b^{27/3} = b^9 \]. This is much simpler to work with and can be directly evaluated.
Algebraic Simplification
Algebraic simplification involves breaking down complex expressions into simpler forms. By using rules of exponents and roots, we make the expressions manageable. The steps generally include:
  • Converting roots to fractional exponents.
  • Simplifying the fractional exponents.
  • Combining like terms.

Starting with \(\sqrt[5]{a^{10}}\), convert it to \[ a^{10/5} = a^2 \]. For \(\sqrt[3]{b^{27}}\), convert it to \[ b^{27/3} = b^9 \]. Each step reduces the complexity of the expressions.

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, solve. (a) \(\sqrt{u}+1=\sqrt{u+4}\) (b) \(\sqrt{n-5}+4=\sqrt{3 n+7}\)

In the following exercises, solve. \(\sqrt{a}+1=\sqrt{a+5}\)

In the following exercises, solve. \(\sqrt{5 v-2}+5=0\)

In the following exercises, simplify by rationalizing the denominator. $$ \frac{\sqrt{150 x^{2} y^{6}}}{\sqrt{6 x^{4} y^{2}}} $$

In the following exercises, simplify by rationalizing the denominator. $$ \frac{3}{5 \sqrt{8}} $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

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.