/*! 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 69 Simplify. $$\left(3 x^{2} y\ri... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 69

Simplify. $$\left(3 x^{2} y\right)\left(2 x^{2} y^{2}\right)^{3}$$

Short Answer

Expert verified
The simplified form of the given expression is \(24x^{8}y^{7}\).

Step by step solution

01

Expand the Second Term

To expand \((2 x^{2} y^{2})^{3}\), apply the power of a power rule which states that \( (a^{m})^{n}=a^{m n}\). Therefore, it becomes \(2^{3} (x^{2})^{3} (y^{2})^{3}\) which equals \(8x^{6}y^{6}\). The whole expression then reads as \(3 x^{2} y * 8x^{6}y^{6}\).
02

Simplify the Expression

To simplify \(3 x^{2} y * 8x^{6}y^{6}\), apply the rule of multiplying like bases which states that \( a^{m}*a^{n}=a^{m+n}\). Therefore, combine the like terms and the expression simplifies to \(24x^{8}y^{7}\).

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.

Exponents
Exponents are a shorthand way of expressing repeated multiplication of the same number or variable. For example, when we write \( x^n \), it means \( x \) is multiplied by itself \( n \) times. So, \( x^3 \) equals \( x \times x \times x \). Exponents have several important rules that help to simplify expressions. Here are a few key points:
  • You add exponents when you multiply terms with the same base. For example, \( a^m \times a^n = a^{m+n} \).
  • You multiply exponents when you raise a power to another power, like \( (a^m)^n = a^{m \cdot n} \).
  • Any number raised to the power of zero equals one, that is, \( a^0 = 1 \), assuming \( a eq 0 \).
Understanding these rules makes it quicker and easier to handle complex exponential terms in polynomial expressions.
Power of a Power Rule
When dealing with the power of a power, you are applying exponents multiple times on the same base. Consider the expression \( (a^m)^n \). This rule tells us to multiply the exponents: \( a^{m \cdot n} \). This makes any expression inside the parentheses simplified to one single exponent outside.

For instance, if you have \( (2x^2y^2)^3 \) like we did in the exercise, you raise each element inside to the third power:
  • \( 2^3 \) becomes 8.
  • \( (x^2)^3 \) becomes \( x^{2 \cdot 3} = x^6 \).
  • \( (y^2)^3 \) becomes \( y^{2 \cdot 3} = y^6 \).
Each part of the term undergoes the same transformation using the power of a power rule. It simplifies larger expressions into more manageable ones.
Multiplying Like Bases
Once each element of your expression is simplified, you can combine terms that have the same base. This process is referred to as multiplying like bases. The rule states: \( a^m \times a^n = a^{m+n} \). This means you add the exponents together when the base is the same.

Returning to our exercise, after simplifying \( 3 x^2 y \times 8 x^6 y^6 \), where you:
  • Multiply the coefficients 3 and 8 to get 24.
  • Add the exponents of \( x \), which are 2 and 6, giving \( x^{8} \).
  • Add the exponents of \( y \), which are 1 and 6, giving \( y^7 \).
The expression reduces to \( 24x^8y^7 \). This rule helps in compacting and simplifying expressions, making calculations faster and more efficient.

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 expression \(w(3 w-1) \mathrm{cm}^{2}\) represents the area of a rectangle of width \(w\) Describe in words the relationship between the length and width of the rectangle.

The frequency (in oscillations per second) of a microwave generated by a microwave oven is approximately \(2,450,000,000\) hertz. (One hertz is one oscillation in 1 s.) Write this number in scientific notation.

If \(x\) is a nonzero real number, is \(x^{-3}\) always positive, always negative, or positive or negative depending on whether \(x\) is positive or negative? Explain your answer.

Multiply. $$(4-3 y)(4+3 y)$$

Find \(\left(4 n^{3}\right)^{2}\) if \(2 n-3=4 n-7\)
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Applied 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.