/*! 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 49 Rewrite each expression as simpl... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 49

Rewrite each expression as simply as you can. $$\left(-m^{2} n\right)^{4}$$

Short Answer

Expert verified
Simplifies to \(m^{8} n^{4}\).

Step by step solution

01

Apply the power rule to each component

The power rule states that \(a^{m}\big)^{n} = a^{m \times n}\). Apply this rule to each component of the expression \((-m^{2} n)^{4}\).
02

Simplify the base

We have two parts in the base: \(m^{2}\) and \(-n\). We need to apply the exponent of 4 to both parts.For the term \(m^{2}\), apply the power rule: \((m^{2})^{4} = m^{2 \times 4} = m^{8}\).For the term \(-n\), the power applies as follows: \((-n)^{4}\) which equals \(n^{4}\) because raising a negative number to an even power makes it positive.
03

Combine the simplified components

Combine the results from Step 2: \(m^{8} \times n^{4}\).Therefore, \((-m^{2} n)^{4}\) simplifies to \(m^{8} n^{4}\).

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.

Power Rule
When working with exponents, the power rule is incredibly useful for simplifying expressions. The power rule states that for any base\(a\) and integers\(m\)and\(n\), the expression\((a^m)^n\) can be simplified to\(a^{m \times n}\).
Let's break this down:
  • We start with an expression where an exponent is raised to another exponent.
  • Instead of manually multiplying the base multiple times, we can just multiply the exponents together.
In our exercise, we applied the power rule to\( (-m^2 n)^4 \). Each component within the base was raised to the fourth power, transforming our expression.
This simplification method saves us time and helps to manage more complex problems with ease.
Base and Exponent
Understanding the relationship between the base and exponent is key in working with exponents. The base is the number being multiplied, and the exponent tells us how many times to multiply the base by itself.
For example, in\(5^3\), 5 is the base and 3 is the exponent. This means we multiply 5 by itself three times:\(5 \times 5 \times 5 = 125\).
In our exercise, we have a combined base of\( -m^2 n \) and the exponent 4.
  • The base\( -m^2 n \) is composed of both a variable with an exponent and another variable without an exponent.
  • By applying the exponent of 4 to both parts of the base, we ensure each part is considered in our final simplified form.
Understanding this relationship helps avoid mistakes and leads to accurate simplifications.
Properties of Exponents
By mastering the properties of exponents, you can simplify even the most complex expressions. Some key properties include:
  • **Product of Powers Property:**\(a^m \times a^n = a^{m+n}\)
  • **Quotient of Powers Property:**\( \frac{a^m}{a^n} = a^{m-n}\)
  • **Power of a Power Property:**\((a^m)^n = a^{m \times n}\)
    • These properties allow you to manipulate and simplify expressions efficiently. In our exercise, we specifically used the power of a power property. When we see\( (m^2)^4 \), this property tells us it can be simplified to\( m^{2 \times 4} = m^8 \).
    Similarly, with\( (-n)^4 \), since raising a negative number to an even power results in a positive number, we get\( n^4 \).
    By combining these simplified components:
    • From\( (m^2)^4 \), we get\( m^8 \)
    • From\( (-n)^4 \), we get\( n^4 \)
    Our final simplified expression\( m^8 n^4 \) showcases how powerful these properties are in streamlining our calculations.

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

There are six ways to pair these four equations. i. \(y=2 x+4\) ii. \(y+2 x=-4\) iii. \(x=4-\frac{y}{2}\) iv. \(2 y-4 x=10\) a. Predict which pairs of equations do not have a common solution. b. Verify your results for Part a by carefully graphing both equations in each pair you selected. Explain how the graphs do or do not verify your prediction. c. Predict which pairs, if any, have a common solution. d. Verify your results for Part c by graphing both equations in each pair you selected. Explain how the graphs do or do not verify your prediction. e. Use your graphs to find a common solution of each pair of equations you listed in Part c.

Recall that for linear equations, first differences are constant; and that for quadratic equations, second differences are constant. Determine whether the relationship in each table could be linear, quadratic, or neither. $$ \begin{array}{|r|r|}\hline x & {y} \\ \hline-3 & {-4} \\ {-2} & {1} \\ {-1} & {4} \\ {0} & {5} \\ {1} & {4} \\ {2} & {1} \\ \hline\end{array} $$

Identify the three pairs of equivalent equations. $$ \begin{array}{ll}{\text { a. } p=2 q-4} & {\text { b. } p-2 q=4} \\ {\text { c. } p-2 q-4=0} & {\text { d. }-2 p=8+4 q} \\ {\text { e. }^{-} p-4=2 q} & {\text { f. } 0.5 p=q-2}\end{array} $$

Tamika and Lydia were making hair ribbons to sell at a crafts fair. Tamika cut seven segments from one length of ribbon and had 2 feet left over. Lydia said, "T m cutting segments twice as long as yours. If your length of ribbon had been just 1 foot longer, I could have cut four segments from it." From their conversation, determine how long Tamika's and Lydia's segments were.

List five values that satisfy each inequality. Include negative and positive values, if possible. $$0 \leq|b| \leq 6$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Probability and Statistics

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.