Simplify and write the resulting expression with only positive exponents. (4 x^2 y)(2 x y^3)^2

Name: Problem 32, Chapter 0: Algebra and Trigonometry
Uploaded: 2021-06-13T19:55:35-08:00
Duration: 4 min 57 s
Channel: Michael Cooper
Description: Simplify and write the resulting expression with only positive exponents. (4 x^2 y)(2 x y^3)^2

Simplify and write the resulting expression with only...

Key Concepts

Simplifying_Algebraic_Expressions

This refers to the process of reducing an algebraic expression to its simplest form. It involves performing arithmetic operations on numerical coefficients, combining like terms, and applying exponent laws. Simplification makes complex expressions easier to manage and is fundamental in algebra.

Expressing_with_Positive_Exponents

Standard form in algebra often requires expressions to have only positive exponents. If a simplified expression includes negative exponents, they are rewritten as reciprocals to conform to this standard. This process ensures clarity and consistency in mathematical communication.

Laws_of_Exponents

This concept involves rules such as the product of powers rule and the power of a product rule. The product of powers rule states that when multiplying like bases, you add the exponents. The power of a product rule indicates that when a product is raised to an exponent, each factor in the product is raised to that exponent. These rules are essential for simplifying expressions involving exponents.

Transcript

00:01 In this exercise, we're being asked to evaluate and simplify an exponential expression that's fairly complicated, but we're going to break it down into steps.

00:10 We first want to look at the entire expression through the thinking about order of operations and see what i noticed.

00:20 The first thing i noticed, i've got two set of parentheses, but i can't really simplify what is inside the parentheses.

00:25 However, i want to start with parentheses and see if there are any exponents.

00:31 Or multiplication or division that i can do with the sets of parentheses.

00:36 And i noticed that i've got an exponent here, the two.

00:42 So i am going to apply that exponent to this set of parentheses.

00:48 I've got a product of three factors inside that parentheses, and that's being raised the second power.

00:55 So that's my first step.

00:57 I am going to use the power to, or the product to power property, exponents to rewrite that set of parentheses.

01:06 I don't want to forget about the rest of my expressions.

01:10 I'm going to write all of the pieces, only rewriting the parts that i can according to order of operations using the product power rule.

01:22 Product power rule says that all three of these factors are going to get that power of two.

01:28 So this becomes two to the second power times x to the second power times y to the third power, but that y to the third power is being raised to the second power.

01:47 So i am applying this exponent of two to all three of the factors in my product in that set of parentheses.

02:05 Then i go back to parentheses in my order of operations.

02:10 I go back to the first step and see, are there any parentheses i can do anything with? i can't do anything with the 4 x squared y yet.

02:20 That is as simple as it gets.

02:24 2 squared is 4, so i can rewrite that.

02:30 I can't rewrite x squared.

02:33 Y to the third power to the second power is the power rule for exponents.

02:38 That says that i multiply the exponents...