/*! 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 22 Simplify.$$\left(x^{2}\right)^{9... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 22

Simplify.$$\left(x^{2}\right)^{9} \cdot\left(x^{5}\right)^{3}$$

Short Answer

Expert verified
The simplified expression is \(x^{33}\).

Step by step solution

01

Apply the Power of a Power Rule

Start by simplifying each part of the expression using the power of a power rule. The rule is \((a^m)^n = a^{m \cdot n}\). For \((x^2)^9\), apply the rule to get \(x^{2 \cdot 9} = x^{18}\). For \((x^5)^3\), apply the rule to get \(x^{5 \cdot 3} = x^{15}\).
02

Multiply the Simplified Expressions

Now that you have \(x^{18}\) from \((x^2)^9\) and \(x^{15}\) from \((x^5)^3\), multiply these together. When multiplying expressions with the same base, you add the exponents: \(x^{18} \cdot x^{15} = x^{18+15}\).
03

Add the Exponents

Add the exponents together: \(18 + 15 = 33\). Therefore, \(x^{18} \cdot x^{15} = x^{33}\). The expression simplifies to \(x^{33}\).

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.

Power of a Power Rule
When you encounter an expression where a base is raised to an exponent, and then that entire expression is raised to another exponent, you can use the Power of a Power Rule. This rule states that \[(a^m)^n = a^{m \cdot n}\]The process involves multiplying the exponents together, keeping the base same.
This is a crucial rule because it simplifies the expression by reducing the number of operations.
  • Consider \((x^2)^9\). Here, the base is \(x\), and you face two exponents: 2 and 9.
  • According to the Power of a Power Rule, multiply these exponents: \(2 \cdot 9 = 18\).
  • Thus, \((x^2)^9\) simplifies to \(x^{18}\).
It's like a shortcut to handle complex expressions! Practice this rule to make exponentiation problems a lot easier.
Multiplying Exponents
Once you've simplified expressions using the power of a power rule, the next step is often to multiply these expressions together. This involves multiplying powers that have the same base.
This is where the rule of adding exponents comes into play.
  • When you multiply terms like \(x^{18} \cdot x^{15}\), keep the common base \(x\).
  • Then add the exponents: \(18 + 15\).
  • The result is \(x^{33}\).
The key takeaway here is that multiplying similar bases involves exponent addition. It simplifies the multiplication of numerous variables into a more manageable form.
By mastering how to handle and manipulate these expressions, you鈥檒l grow more comfortable with algebraic operations, seeing them as straightforward tasks rather than complex hurdles.
Simplifying Expressions
Simplifying expressions is all about making complex algebraic terms more manageable. With exponentiation, this step often involves applying rules like the power of a power rule and the rule for multiplying exponents.
By breaking down problems into these smaller parts, you simplify them step-by-step.
  • First, apply the rules you know to the expression: \((x^2)^9 \cdot (x^5)^3\).
  • By simplifying, you get \(x^{18}\) and \(x^{15}\), which then multiply together to \(x^{33}\).
  • This final result is much simpler than the original more complex expression.
These simplifications make the operations not only easier to handle mathematically but also more elegant.
Remember, simplifying is about finding efficiency and clarity in mathematics. It equips you to solve more challenging problems by breaking them down into simpler components.

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. Write \(\frac{3}{8}\) as a percent.

Multiply. $$7(x-3)$$

Multiply. $$(2 x-6)(x+4)$$

Multiply. $$-3(8 r+3 s)$$

Multiply. $$(y+7)(y-7)$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

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.