/*! 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 95 Simplifying \(\left(a^{m}\right)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 95

Simplifying \(\left(a^{m}\right)^{n}\) $$ \left(x^{4}\right)^{3} $$

Short Answer

Expert verified
The simplified form of \((x^4)^3\) is \x^{12}\.

Step by step solution

01

Understand the Exponentiation Rule

When simplifying the expression \((a^m)^n\), use the power of a power rule, which states \(a^{m \times n}\). This means that you multiply the exponents.
02

Apply the Rule to the Given Expression

Apply the power of a power rule to \((x^4)^3\). According to the rule, this becomes \(x^{4 \times 3}\).
03

Simplify the Exponents

Now, multiply the exponents: \(4 \times 3 = 12\). So, \((x^4)^3 = x^{12}\).

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
The power of a power rule is one of the key concepts in exponentiation. This rule helps us simplify expressions where an exponentiated number is raised to another exponent. To use this rule, you multiply the exponents. For example, when you have \( (a^m)^n \), you apply the rule to get \(a^{m \times n}\).
This simplifies the work significantly and reduces potential errors when dealing with larger expressions.
In the given problem, \( (x^4)^3 \), you apply the power of a power rule as follows:
  • First, recognize the base and the exponents. Here, the base is \(x\), the first exponent is \(4 \), and the second exponent is \3 \.
  • Then, multiply these exponents: \( 4 \times 3 \).
  • This gives \(x^{12} \), a much simpler expression.
Understanding this rule lays the groundwork for more complex exponentiation problems.
Multiplying Exponents
Another fundamental concept is multiplying exponents. This principle is crucial for understanding how to deal with expressions that involve multiple layers of exponents. The power of a power rule directly relies on this multiplication.
If we look at the problem again, we started with \( (x^4)^3 \). To simplify it, we needed to multiply the exponents:
\[ 4 \times 3 = 12 \]
This multiplication transforms the expression into \ x^{12} \.
Here are some tips to make this process smoother:
  • Always identify each exponent clearly.
  • Use parentheses to keep clarity around different parts of the expression.
  • Remember that multiplying the exponents is different from adding them, which occurs in other types of exponentiation problems.
Practice this with various exercises to become more confident with these operations.
Simplifying Expressions
Simplifying expressions is a crucial skill in algebra that makes complex equations easier to work with. By applying rules like the power of a power rule and multiplying exponents, we can transform and simplify expressions to their most basic form.
In our example, \( (x^4)^3 \) became \(x^{12}\) through simplification steps:
  • First, we identified the exponents involved.
  • Next, we applied the power of a power rule by multiplying the exponents.
  • Finally, we simplified the expression to \(x^{12} \).
Always remember: simplify expressions step-by-step. Don't rush. Each rule has its role, and understanding these roles helps in solving algebraic problems efficiently.
Practicing these simplification techniques with different problems will build a strong foundation in algebra.

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

Use roster notation to write each set. The set of letters in the word "algebra"

To the student and the instructor: Throughout this text selected exercises are marked with the icon Ahal. Students who pause to inspect an Ahal exercise should find the answer more readily than those who proceed mechanically. This may involve looking at an earlier exercise or example, or performing calculations in a more efficient manner. Some Ahal exercises are left unmarked to encourage students to always pause before working a problem. Evaluate each expression using the values provided. \([5(r+s)]^{2},\) for \(r=1\) and \(s=2\)

Use set-builder notation to write each set. The set of all multiples of 5 between 7 and 79

Find an equivalent expression by factoring. $$ 5 x+50 $$

Calculate using the rules for order of operations. If an expression is undefined, state this. $$ \frac{3^{4}-(5-3)^{4}}{8-2^{3}} $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

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.