/*! 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 53 Simplify. \(\left(-3 x^{2} y^{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 53

Simplify. \(\left(-3 x^{2} y^{6}\right)^{0}\)

Short Answer

Expert verified
1

Step by step solution

01

- Understand the Zero Exponent Rule

Any non-zero base raised to the power of 0 is equal to 1. This is known as the zero exponent rule. Mathematically, this is expressed as: \[ a^0 = 1 \] for any non-zero value of \(a\).
02

- Apply the Zero Exponent Rule

In the given expression, \( (-3 x^2 y^6)^0 \), the entire base is raised to the power of 0. According to the zero exponent rule, we can simplify this to: \[ (-3 x^2 y^6)^0 = 1 \]

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.

Exponentiation
Exponentiation is a mathematical operation involving numbers known as bases and exponents. The exponent tells us how many times to multiply the base by itself. For example, in the expression \(a^n\), \(a\) is the base and \(n\) is the exponent. Here are some key points:
  • \(a^1\) means the base is used once: \(a \).
  • \(a^n\) means the base is multiplied by itself \(n\) times, e.g., \(2^3 = 2 \cdot 2 \cdot 2 = 8\).
  • An exponent of 1 means the number is unchanged.
The concept of exponentiation is critical for understanding more advanced algebraic expressions. In the given exercise \left(-3 x^{2} y^{6}\right)^{0}\, we need to apply our understanding of exponents to simplify the expression.
Simplification
Simplification in mathematics means breaking down complex expressions into their simplest form for easier interpretation and solving. Here are some simple guidelines to help:
  • Combine like terms where possible. For example, \(3x + 4x = 7x\).
  • Use algebraic properties such as the distributive property to simplify.
  • Follow the order of operations (PEMDAS/BODMAS) to avoid mistakes.
In our specific exercise, we use the zero exponent rule for simplification. Understanding how to simplify is fundamental to solving algebraic expressions efficiently.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols. Examples include:
  • \(2x + 3\)
  • \(x^2 - 4y + 7\)
Algebra uses letters for variables to represent numbers. These can change based on the situation. More complex expressions may involve multiple variables and higher exponents. For the given exercise, \((-3x^2y^6)^0\), the entire expression is inside the parentheses. Knowing how to interpret and manipulate these expressions is vital for understanding algebra.
Zero Exponent
The zero exponent rule is a specific property in mathematics stating that any non-zero base raised to the power of 0 is equal to 1. Consider the following examples:
  • \(5^0 = 1\)
  • \((-7)^0 = 1\)
  • \((-3x^2y^6)^0 = 1\)
The intuition behind this rule is related to the idea of repeated multiplication. Because the exponent tells you how many times to use the base as a factor, using it zero times essentially gives you the multiplicative identity, which is 1. This concept is fundamental when dealing with algebraic expressions involving exponents.

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

Factor completely. $$ n^{6}-1 $$

Working together, Katherine and Julianna can plant new trees on their recently reforested land in 6 days. Working alone, it would take Julianna 2 days longer than it would take Katherine to plant the trees. How long would it take Katherine, working alone, to plant the trees?

Find an equation in slope鈥搃ntercept form of a line with the given characteristics. Slope: \(2 ;\) contains \((-3,7)\)

Number of Handshakes. There are \(n\) people in a room. The number \(N\) of possible handshakes by the people is given by the function $$N(n)=\frac{n(n-1)}{2}$$ For what number of people \(n\) is \(66 \leq N \leq 300 ?\)

Sump Pump. The lift distances for a Liberty 250 sump pump moving fluid at various flow rates are shown in the following table. \(\begin{array}{|c|c|}\hline \text { Gallons per } & {\text { Lift Distance }} \\\ {\text { Minute }} & {\text { (in feet) }} \\ \hline 10 & {21} \\ {20} & {18} \\ {40} & {8} \\ \hline\end{array}\) a) Let \(x\) represent the flow rate, in gallons per minute, and \(d(x)\) the lift distance, in feet. Find a quadratic function that fits the data. b) Use the function to find the lift distance for a flow rate of 50 gal per min.
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

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.