/*! 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 45 Raise each monomial to the indic... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 45

Raise each monomial to the indicated power. $$\left(2 a^{2} b^{3}\right)^{6}$$

Short Answer

Expert verified
\((2a^2b^3)^6 = 64a^{12}b^{18}\).

Step by step solution

01

Understanding the Problem

We need to raise the monomial \(2a^2b^3\) to the power of 6. This involves applying the power to each factor in the monomial.
02

Applying the Power of a Product Rule

The power of a product can be distributed to each factor in the product. This means \((xy)^n = x^n y^n\). So, \((2a^2b^3)^6 = 2^6 (a^2)^6 (b^3)^6\).
03

Simplifying Each Factor

Now, simplify each factor using the rules of exponents: - \(2^6 = 64\)- \((a^2)^6 = a^{2 \times 6} = a^{12}\)- \((b^3)^6 = b^{3 \times 6} = b^{18}\)
04

Writing the Final Expression

Combine the simplified factors to write the final expression: \((2a^2b^3)^6 = 64a^{12}b^{18}\).

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.

Monomial
A monomial is a mathematical expression consisting of a single term which can be a constant, a variable, or the product of constants and variables. In our exercise, the monomial is \(2a^2b^3\). Here, we have:
  • Coefficient: The constant number in front of the variables, which is 2 in this case.
  • Variables: These are "\(a\)" and "\(b\)" that are raised to powers.
  • Exponents: The powers to which the variables are raised, "\(a^2\)" means "a squared," and "b^3" means "b cubed."
Understanding monomials is vital because they are the building blocks of more complex polynomials. It is essential to master how to manipulate these expressions under various mathematical operations like multiplication, division, and exponentiation.
Power of a Product
The 'power of a product' is a crucial concept particularly useful when raising monomials that consist of multiple factors to a power. In our example, we need to compute \((2a^2b^3)^6\). The power of a product states that when a product is raised to an exponent, each factor in the product is raised to that exponent. This can be easily expressed as:\[ (xy)^n = x^n y^n \]For our exercise, this rule transforms our expression into \(2^6 (a^2)^6 (b^3)^6\). It allows us to distribute the exponent over each factor separately, making it much easier to handle complex expressions. Remember, the power applies not just to variables but to any product of numbers and/or variables.
Exponent Rules
Exponent rules are fundamental to solving problems involving powers. They provide a set of guidelines to simplify expressions efficiently. Let's apply these rules to our exercise step-by-step:
  • Power of a Power Rule: This rule states that when raising a power to another power, you multiply the exponents. It is represented by \((x^m)^n = x^{m \,* n}\). For example, \((a^2)^6 = a^{12}\) and \((b^3)^6 = b^{18}\).
  • Power of a Product Rule: We already applied this in the previous section, allowing us to handle expressions like \((2a^2b^3)^6\) by raising each component to the 6th power.
By applying these rules, we simplify \(2^6 = 64\), \(a^{12}\), and \(b^{18}\), leading us to the final expression of \(64a^{12}b^{18}\). These rules make it efficient to handle even more complex expressions systematically.

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

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$w^{2}-4 w=5$$

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$3 x^{4}-81 x$$

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$25 n^{2}+64$$

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$15 x^{2}+34 x+15=0$$

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$4 x^{2}+16$$
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation

Calculus

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.