/*! 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 35 Simplify each of the following a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 35

Simplify each of the following as completely as possible. $$\left(-x^{3} y\right)^{3}$$

Short Answer

Expert verified
The simplified form is \鈭抶^{9}y^{3}\.

Step by step solution

01

- Understand the Expression

Identify the given expression \((鈭抶^{3} y)^{3}\). The goal is to simplify this expression.
02

- Apply the Power Rule

Use the power rule \((a \times b)^m = a^m \times b^m\) to separate the powers of each factor inside the parentheses. This gives \((鈭抶^{3} y)^{3} = (鈭�1)^{3} \times (x^{3})^{3} \times (y)^{3}\).
03

- Simplify Each Term Individually

Calculate each part: \1. \((鈭�1)^{3} = -1 \)2. \((x^{3})^{3} = x^{3 \times 3} = x^{9} \) 3. \((y)^{3} = y^{3}\)
04

- Combine the Simplified Terms

Combine the simplified terms from Step 3 to get the final answer: \(-1 \times x^{9} \times y^{3}\). Simplify this to \鈭抶^{9}y^{3}\.

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 in Algebra
The power rule in algebra is a fundamental technique for simplifying expressions that involve exponents.
It states that \[ (a \times b)^m = a^m \times b^m \] where 'a' and 'b' are any numbers or algebraic expressions, and 'm' is an exponent.
When you apply this rule, you distribute the exponent 'm' to both 'a' and 'b'.
For example, \[ (2xy)^3 = 2^3 \times x^3 \times y^3 = 8x^3y^3. \]
In the provided exercise, using this rule helps simplify \[ (-x^3y)^3 = (-1)^3 \times (x^3)^3 \times (y)^3. \] The power rule is especially useful when dealing with complex expressions because it breaks them down into simpler, more manageable parts.
Simplifying Exponents
Simplifying exponents involves applying specific rules to make expressions more straightforward.
One key rule is the power of a power rule: \[ (x^a)^b = x^{a \times b}. \] This means when an exponent is raised to another exponent, you multiply the exponents.
For instance, \[ (x^3)^2 = x^{3 \times 2} = x^6. \]
In the exercise, we applied this rule as follows: \[ (x^3)^3 = x^{3 \times 3} = x^9. \]
Another technique is handling negative bases, where you preserve the sign of the base and raise it to the power of the exponent.
Thus, \[ (-1)^3 = -1. \] Simplifying exponents helps to make the final algebraic expression compact and easier to interpret.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations.
They form the foundation of algebra and are crucial in representing mathematical relationships.
For example, the expression \[ -x^9y^3 \] is composed of the variable 'x' raised to the power of 9, multiplied by the variable 'y' raised to the power of 3, and a coefficient of -1.
Simplifying algebraic expressions involves using rules like the power rule and exponent rules to make them as simple as possible.
This process often includes:
  • Combining like terms
  • Applying powers and roots
  • Simplifying fractions
Understanding these expressions allows for further mathematical operations and problem-solving tasks, essential for mastering algebra.

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

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$\left(x^{3}+y^{2}\right)^{2}$$

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(3 x-4)(5 x+7)$$

Convert each number into scientific notation. $$28.40 \times 10^{6}$$

Do your computations using scientific notation. The diameter of an atom might typically be 3.47 A. Write this number in scientific notation.

Do your computation using scientific notation. $$\frac{(480)(0.008)}{(0.24)(4000)}$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Decision Maths

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.