/*! 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 65 Assume that all variables repres... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 65

Assume that all variables represent positive real numbers. $$ \sqrt{y^{12}} $$

Short Answer

Expert verified
\( y^6 \)

Step by step solution

01

Identify the Power Under the Square Root

The expression is given as \( \sqrt{y^{12}} \). Notice that there is a power of 12 under the square root.
02

Apply the Power of a Power Rule

Recall that \( \sqrt{x} \) is equivalent to \( x^{1/2} \). Therefore, \( \sqrt{y^{12}} = (y^{12})^{1/2} \).
03

Simplify the Expression Using Exponents

Use the rule \((a^m)^n = a^{m \times n}\). Here, \((y^{12})^{1/2} = y^{12 \times (1/2)} = y^{6}\).
04

Final Answer

The simplified expression of \( \sqrt{y^{12}} \) is \( y^6 \).

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.

Exponent Rules
Exponent rules are fundamental when it comes to simplifying expressions with variables raised to powers. Understanding these rules makes it easier to manipulate expressions and solve problems. One primary rule used in the problem above is the power of a power rule. This rule states that when you have an exponent raised to another exponent, you multiply the exponents together:
  • If you have \((a^m)^n\), it simplifies to \(a^{m \times n}\).
In our exercise, this rule is applied by taking \((y^{12})^{1/2}\) and simplifying to \(y^{12 \times (1/2)}\), which results in \(y^{6}\). Remembering these rules helps to quickly reduce complex expressions in mathematics.
Square Roots
Square roots are another crucial concept when working with radical expressions. The square root of a number or an expression is a value that, when multiplied by itself, gives the original number or expression.
  • The notation \(\sqrt{x}\) is equivalent to \(x^{1/2}\).
This indicates that for any expression inside the square root, you are effectively raising that expression to the power of \(1/2\). Considering this definition helps simplify expressions under a square root, especially when combined with exponent rules. In the example of \(\sqrt{y^{12}}\), recognizing \(\sqrt{y^{12}} = (y^{12})^{1/2}\) allows us to use exponent rules to further simplify it.
Positive Real Numbers
In mathematics, assuming all variables are positive real numbers can simplify calculations and interpretations significantly. Positive real numbers are all the numbers greater than zero and are not imaginary. This assumption can make algebraic manipulations, such as square roots, much easier since we avoid complications introduced by negative numbers or variables.
  • For positive real numbers, the square root operation is straightforward and does not result in complex numbers.
In the problem we solved, recognizing that \(y\) is a positive real variable means that taking \(\sqrt{y^{12}}\) simplifies to a real number without additional constraints or considerations. The simplification to \(y^{6}\) becomes direct given that exponent rules neatly apply without the need for further clarifications.

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

Solve. $$ \sqrt{3 y+6}=\sqrt{7 y-6} $$

Consider the equations \(\sqrt{2 x}=4\) and \(\sqrt[3]{2 x}=4\) a. Explain the difference in solving these equations. b. Explain the similarity in solving these equations.

Rationalize each numerator. Assume that all variables represent positive real numbers. \(\sqrt{\frac{18 x^{4} y^{6}}{3 z}}\)

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

Simplify. $$ \frac{\frac{1}{y}+\frac{4}{5}}{\frac{-3}{20}} $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

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.