Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Which of the following expressions is equivalent to \(\left(-2 x^{5} y^{2}\right)^{4} ?\) A. \(-16 x^{20} y^{8}\) B. \(-8 x^{20} y^{8}\) C. \(-8 x^{9} y^{6}\) D. 16\(x^{9} y^{6}\) E. 16\(x^{20} y^{8}\)

Short Answer

Expert verified
Answer: 16\(x^{20} y^{8}\)

Step by step solution

01

Raise components to the power of 4

Applying the rule \((a b)^n = a^n b^n\), we can raise the components (-2), \(x^5\), and \(y^2\) to the power of 4: \(\left(-2 x^{5} y^{2}\right)^{4} = (-2)^4 (x^5)^4 (y^2)^4\)
02

Evaluate each component

Calculate the value of each component after raising them to the power of 4: \((-2)^4 = 16\) \((x^5)^4 = x^{5 \cdot 4} = x^{20}\) (Using rule \((a^n)^m = a^{n \cdot m}\)) \((y^2)^4 = y^{2 \cdot 4} = y^8\)
03

Combine the results

Now we can combine the results : \(16 x^{20} y^8\) Looking at the answer choices, we can see that this expression matches option E. So, the correct answer is: E. 16\(x^{20} y^{8}\)

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 Vaia!

Key Concepts

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

ACT Math
In the realm of standardised tests, ACT Math stands tall as a critical component purposed to assess a student’s proficiency in mathematical reasoning and application of concepts learned through high school. The questions span various topics, including algebra, geometry, and trigonometry, to name a few.

For students vying to conquer the ACT Math section, mastering a multitude of skills is imperative. One such skill lies in polynomial simplification, which often requires fluency in exponent rules and manipulation of algebraic expressions. The ability to simplify polynomials can make a significant difference in a student's performance, not only in the ACT but also in their academic pursuits in mathematics-related courses.

As with our example problem, understanding and correctly applying exponent rules when simplifying polynomials can directly impact the success rate of solving similar questions typically found in the ACT Math section. Accurate simplification leads to identifying the correct choice among multiple answer selections, a skill that echoes the real-time decision-making required during the test.
Exponent Rules
Exponent rules, often referred to as the laws of exponents, are foundational for manipulating and simplifying expressions that contain powers. These rules govern the operations performed on exponential terms and are essential for solving a variety of algebraic problems.

Key Rules of Exponents

  • \textbf{Product of Powers}: For bases that are the same, \(a^{m} \times a^{n} = a^{m+n}\text{,}\) you add the exponents.
  • \textbf{Power of a Power}: When raising an exponent to another power, like \(a^{n})^{m} = a^{n \times m},\) you multiply the exponents.
  • \textbf{Power of a Product}: The rule \(\(ab\)^{n} = a^{n}b^{n}\) tells us to raise each part of the product to the power.
  • \textbf{Negative Exponent}: For \(a^{-n} = \frac{1}{a^{n}},\) the negative exponent indicates a reciprocal.
  • \textbf{Zero Exponent}: Any number with a zero exponent equals one, as in \(a^{0} = 1.\)

In practice, the correct usage of these laws allows for the simplification of complex expressions, such as \(\left(-2 x^{5}y^{2}\right)^{4},\) into a more manageable form like \(16 x^{20} y^8.\) Missteps in the application can lead to incorrect solutions, so a firm grasp of the exponent rules is a non-negotiable asset for students.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operation signs. They represent quantities in symbolic form and allow for general calculations that apply to various values. Students encounter algebraic expressions early in their math education, and these expressions become increasingly complex with advancement in curriculum levels.

Working with algebraic expressions, such as polynomial terms in our example \(\left(-2 x^{5}y^{2}\right)^{4},\) demands proficiency in combining like terms, applying the distributive property, and manipulating variables according to exponent rules. Simplifying algebraic expressions reduces them to the simplest form, making it easier to understand, interpret and, notably, solve for unknown variables.

Mathematical fluency in expressions is not solely beneficial for exercises; it's fundamentally intertwined with real-world problem-solving. Whether it's calculating rates of change in physics or plotting functions in computational algorithms, a consequential understandment of algebraic expressions and their simplifications is monumental in a diverse span of academic and professional fields.

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

According to the information provided, a sample that had a calculated \(\delta^{18} \mathrm{O}\) of zero had a \(^{18} \mathrm{O} /^{16} \mathrm{O}\) value that compared in which of the following ways to the 18 \(\mathrm{O} /^{16} \mathrm{O}\) value of the standard sample? The sample's \(\mathrm{O} /^{16} \mathrm{O}\) ratio was: F. \(\frac{1}{2}\) of the \(^{18} 0^{16}\) o ratio of the standard. G. the same as the \(\$ 0 /^{16}\) O ratio of the standard. H. 1\(\frac{1}{2}\) times larger than the \(^{18} 0 \%\) ratio of the standard. J. twice as large as the "O/' O ratio of the standard.

Which of the following findings, if true, would best support the idea that Mars once was "a garden world, a paradise among planets" \(\text { (lines } 15-16) ?\) A. Images of Mars revealed networks of drainage basins and streambeds. B. When the sun was young, it was 25 to 30 percent dimmer than it is today. C. A succession of asteroids and comets struck Mars about every \(10,000\) years. D. Large expanses of ice on Mars were thawed by a heat source deep within the planet.

What are the values for \(x\) that satisfy the equation \((x+a)(x+b)=0 ?\) F. \(-a\) and \(-b\) G. \(-a\) and \(b\) H. \(-a b\) J. \(a\) and \(-b\) K. \(a\) and \(b\)

Based on Figure \(3,\) if \(Y_{0}\) were \(5 \mathrm{cm},\) the fall time on the Moon would be closest to: F. 0.4 sec. G. 0.8 sec. H. 1.9 sec. J. 2.3 sec.

Both scientists would most likely agree that the ocean water was, or became, oxygen-poor when which of the following events occurred? F. Ocean water circulation reversed its usual direction. G. Ocean water circulation slowed, stopped, or was absent. H. Oceanic organisms dramatically increased the available oxygen in the ocean water. J. Oceanic organisms used up all the available \(\mathrm{CO}_{2}\) in the ocean water.

See all solutions

Recommended explanations on English Textbooks

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.

Sign-up for free