Textbook Question
Simplify each exponential expression in Exercises 23–64. (−5x^4 y)(−6x^7 y^11)
827
views
Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 1, Problem 47
In Exercises 39–48, factor the difference of two squares. 16x4−81
Verified step by step guidance1
Recognize that the given expression, 16x^4 - 81, is a difference of two squares. Recall the formula for factoring the difference of two squares: a^2 - b^2 = (a - b)(a + b).
Rewrite each term as a square: 16x^4 can be written as (4x^2)^2, and 81 can be written as 9^2.
Substitute these squares into the formula: (4x^2)^2 - 9^2 = (4x^2 - 9)(4x^2 + 9).
Notice that the first factor, 4x^2 - 9, is itself a difference of two squares. Apply the formula again: 4x^2 - 9 = (2x - 3)(2x + 3).
Combine all the factors: The fully factored form of the expression is (2x - 3)(2x + 3)(4x^2 + 9).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Difference of Squares
The difference of squares is a specific algebraic expression that takes the form a^2 - b^2, which can be factored into (a - b)(a + b). This concept is fundamental in algebra as it simplifies expressions and solves equations efficiently. In the given problem, recognizing that 16x^4 and 81 are both perfect squares allows us to apply this factoring technique.
Recommended video:
06:24
Solving Quadratic Equations by Completing the Square
Perfect Squares
A perfect square is a number that can be expressed as the square of an integer or a variable. For example, 16x^4 is a perfect square because it can be written as (4x^2)^2, and 81 is a perfect square since it equals 9^2. Identifying perfect squares is crucial for factoring expressions involving the difference of squares.
Recommended video:
06:24Solving Quadratic Equations by Completing the Square
Factoring Techniques
Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. Understanding various factoring techniques, including the difference of squares, is essential for solving polynomial equations and simplifying algebraic expressions. Mastery of these techniques enhances problem-solving skills in algebra.
Recommended video:
Guided course
04:36Factor by Grouping
Related Practice
Textbook Question
Evaluate each expression or indicate that the root is not a real number.
623
views
Textbook Question
Determine whether each statement is true or false. -π ≥ - π
1026
views
Textbook Question
Rationalize the denominator.
666
views
Textbook Question
In Exercises 15–58, find each product. (4x2−1)2
1006
views
Textbook Question
Add or subtract as indicated. 2/5x − (x+1)/4x
873
views