/*! 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 Perform the indicated operations... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 35

Perform the indicated operations and simplify. $$ \left(2 x^{2}+3 y^{2}\right)^{2} $$

Short Answer

Expert verified
\(4x^4 + 12x^2y^2 + 9y^4\)

Step by step solution

01

Apply the Binomial Theorem

According to the binomial theorem, we can expand the square of a binomial as follows: \((a + b)^2 = a^2 + 2ab + b^2\). In this case, \(a = 2x^2\) and \(b = 3y^2\).
02

Calculate \(a^2\)

Square the first term, \(a = 2x^2\): \( (2x^2)^2 = 4x^4 \).
03

Calculate \(2ab\)

Multiply the two terms and then multiply by 2: \(2 \cdot (2x^2) \cdot (3y^2) = 12x^2y^2\).
04

Calculate \(b^2\)

Square the second term, \(b = 3y^2\): \((3y^2)^2 = 9y^4\).
05

Combine the Terms

Add all the results from steps 2, 3, and 4 to get the expanded form: \(4x^4 + 12x^2y^2 + 9y^4\). Simplify if necessary.

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.

Binomial Theorem
The Binomial Theorem is a cornerstone of algebra, offering a way to expand expressions raised to a power. In this context, it allows us to expand \( (a + b)^2 \) quickly without manually writing out and multiplying terms by themselves. The theorem states that \( (a + b)^2 = a^2 + 2ab + b^2 \).
  • \(a^2\) is simply the square of the first term.
  • \(b^2\) is the square of the second term.
  • \(2ab\) represents two times the product of the first and second term.
By applying this theorem, complex polynomial expressions become manageable. The key takeaway is that it provides a precise pattern for expansion, which can be generalized to higher powers.
Algebraic Expressions
Algebraic expressions are essentially mathematical phrases that include numbers, variables, and operational signs. Understanding the structure and components of these expressions is crucial for applying algebraic methods effectively.In the given problem, the expression is \( (2x^2 + 3y^2)^2 \). This expression involves:
  • Terms: Components separated by addition or subtraction. Here, \(2x^2\) and \(3y^2\) are terms.
  • Coefficients: Numbers in front of the variables. In \(2x^2\), 2 is the coefficient.
  • Variables and Exponents: Letters representing numbers, and exponents indicating repeated multiplication. \(x^2\) means \(x\) is multiplied by itself.
Recognizing these allows for manipulation and proper application of the binomial theorem to simplify expressions.
Polynomial Simplification
Polynomial simplification involves making an expression easier to work with while still expressing the same mathematical relationship. This process is vital in algebra to reduce complexity and reveal the essential structure of expressions.In the exercise, we simplify the expression \( (2x^2 + 3y^2)^2 \) by using the binomial theorem, resulting in \( 4x^4 + 12x^2y^2 + 9y^4 \). Simplification includes steps such as:
  • Combining like terms: Though not needed here, in some cases terms are combined if they share the same variable factor.
  • Reducing expressions: Breaking terms down or rearranging them for clarity.
Effective simplification not only makes the expression tidier but also prepares it for further algebraic operations.

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

The theory of relativity states that as an object travels with velocity \(v\) , its rest mass \(m_{0}\) changes to a mass \(m\) given by the formula $$m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$$ where \(c \approx 3.0 \times 10^{8} \mathrm{m} / \mathrm{s}\) is the speed of light. By what factor is the rest mass of a spaceship multiplied if the ship travels at one-tenth the speed of light? At one-half the speed of light? At 90\(\%\) of the speed of light? How does the mass of the spaceship change as it travels at a speed very close to the speed of light? Do we need to know the actual value of the speed of light to answer these questions?

How Big Is a Billion? If you have a million \(\left(10^{6}\right)\) dollars in a suitcase, and you spend a thousand \(\left(10^{3}\right)\) dollars each day, how many years would it take you to use all the money? Spending at the same rate, how many years would it take you to empty a suitcase filled with a billion (10^{9} ) dollars?

\(83-88=\) Rationalize the numerator. $$ \frac{\sqrt{3}+\sqrt{5}}{2} $$

Without using a calculator, determine which number is larger in each pair. (a) \(2^{1 / 2}\) or \(2^{1 / 3}\) (b) \(\left(\frac{1}{2}\right)^{1 / 2}\) or \(\left(\frac{1}{2}\right)^{1 / 3}\) (c) \(7^{1 / 4}\) or \(4^{1 / 3}\) (d) \(\sqrt[3]{5}\) or \(\sqrt{3}\)

\(71-76\) m simplify the expression. (This type of expression arises in calculus when using the "quotient rule.") $$ \frac{\left(1-x^{2}\right)^{1 / 2}+x^{2}\left(1-x^{2}\right)^{-1 / 2}}{1-x^{2}} $$
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

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.