/*! 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 14 Find each product. $$ \left(... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 14

Find each product. $$ \left(5 x+\frac{2}{5} y\right)^{2} $$

Short Answer

Expert verified
The product is def 25x^2 + 4xy + \frac{4}{25}y^2.

Step by step solution

01

Understand the Formula

Recall the formula for the square of a binomial: def \[ (a + b)^2 = a^2 + 2ab + b^2 \]
02

Identify Values of a and b

In the expression def \[ (5x + \frac{2}{5}y)^2 \] a = 5x and b = \frac{2}{5}y.
03

Compute a^2

Compute def \[ a^2 = (5x)^2 = 25x^2 \]
04

Compute b^2

Compute def \[ b^2 = (\frac{2}{5}y)^2 = \frac{4}{25}y^2 \]
05

Compute 2ab

Compute def \[ 2ab = 2(5x)(\frac{2}{5}y) = 2 \cdot 5x \cdot \frac{2}{5}y = 4xy \]
06

Combine the Results

Combine all parts to get the final result: def \[ (5x + \frac{2}{5}y)^2 = a^2 + 2ab + b^2 = 25x^2 + 4xy + \frac{4}{25}y^2 \]

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.

square of a binomial
The square of a binomial refers to squaring a two-term algebraic expression, resulting in a three-term expression. Binomials have the form \(a + b\) or \(a - b\). When squared, the formula \((a + b)^2 = a^2 + 2ab + b^2\) is used. Notice how each term is accounted for: \(a^2\) (square of the first term), \(2ab\) (twice the product of the terms), and \(b^2\) (square of the second term). This formula simplifies the multiplication process, making it easier to manage.
algebraic expression
An algebraic expression combines numbers, variables, and arithmetic operations (addition, subtraction, multiplication, division, and exponentiation). In our exercise, \(5x + \frac{2}{5}y\) is an example of a binomial algebraic expression. Such expressions can be manipulated using algebraic rules to simplify or transform them. Understanding the structure aids in applying formulas like the square of a binomial.
polynomial multiplication
Polynomial multiplication involves combining terms from two polynomials to form a new polynomial. For example, squaring the binomial \(5x + \frac{2}{5}y\) entails multiplying it by itself. This process involves distributing each term in the first polynomial across each term in the second. The standard result format, like \(a^2 + 2ab + b^2\), helps systematize and simplify the calculations, ensuring that no terms are missed.
exponentiation
Exponentiation is the operation of raising a quantity to a power. It expresses how many times a number, known as the base, is multiplied by itself. For example, in \( (5x)^2\), \5x\ is the base and 2 is the exponent, signifying \(5x \times 5x.\) Similarly, \left(\frac{2}{5}y\right)^2\ means multiplying \frac{2}{5}y\ by itself: \frac{2}{5}y \times \frac{2}{5}y.\ Exponentiation principles are crucial in simplifying and solving algebraic expressions involving powers.

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

Simplify by writing each expression wth positive exponents. Assume that all variables represent nonzero real numbers. $$ \frac{\left(2 y^{-1} z^{2}\right)^{2}\left(3 y^{-2} z^{-3}\right)^{3}}{\left(y^{3} z^{2}\right)^{-1}} $$

Evaluate. $$ 1000(1.53) $$

Multiply. $$ \frac{1}{2}(4 m-8 n) $$

Evaluate. $$ 100(72.79) $$

Evaluate. $$ 6504 \div 100 $$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Calculus

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Discrete Mathematics

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.