/*! 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 37 Simplify each exponential expres... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 0: Problem 37

Simplify each exponential expression $$ \frac{x^{14}}{x^{-7}} $$

Short Answer

Expert verified
The simplified form of \(\frac{x^{14}}{x^{-7}}\) is \(x^{21}\).

Step by step solution

01

Identify the base and the exponents

In the expression \(\frac{x^{14}}{x^{-7}}\), 'x' is the base and '14' is the exponent of the numerator and '-7' is the exponent of the denominator.
02

Use rule of exponents for division

The law of exponents says that when you divide like bases, you subtract the exponent of the denominator from the exponent of the numerator. So, A^m / A^n = A^(m-n). Using this rule, the expression becomes: \(x^{14-(-7)}\).
03

Simplify the exponent

Subtract -7 from 14 to simplify the exponent. The expression then becomes: \(x^{14+7}\).
04

Final Simplification

Now it's just a basic arithmetic to add the exponents. So the final simplified expression is: \(x^{21}\).

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.

Understanding the Law of Exponents
The law of exponents is a fundamental concept in algebra that provides rules for manipulating exponential expressions. It simplifies the process of working with powers by offering a structured way to combine them. There are several rules, but one of the most important is the division rule. When dividing exponential expressions with the same base, we can subtract the exponents according to the formula:
  • If we have \(A^m \div A^n\), we rewrite it as \(A^{m-n}\).
  • This rule helps to reduce complex expressions into simpler forms.
It's critical to ensure that the base is the same on both expressions you are dividing. If the bases differ, this rule of exponents cannot be applied directly.
Simplification of Exponential Expressions
Simplification in math means making an expression easier to understand or use. For exponential expressions, simplification involves reducing the number of terms and operations by using the laws of exponents.
  • In the given expression \(\frac{x^{14}}{x^{-7}}\), we see that both the numerator and the denominator have the same base \(x\).
  • The exponents can be simplified by applying the relevant rules, leading to a cleaner and more manageable form.
The main aim is to perform operations such as addition or subtraction of exponents where applicable, and to rewrite the expression in its most basic form. This makes it easier for students and professionals to work with these expressions in more complex equations.
Mastering Division of Exponents
The division of exponents is a direct application of the law of exponents. It's particularly useful when managing fractions of powers. In our example, we have \(\frac{x^{14}}{x^{-7}}\).
The steps to solve it involve:
  • Identifying the base \(x\) and the exponents 14 and -7.
  • Using the division rule: subtract the exponent in the denominator from the exponent in the numerator.
  • This transforms our expression to \(x^{14-(-7)}\), which further simplifies to \(x^{14+7}\).
  • Finally, calculate \(14 + 7\) to obtain the simplified exponent of 21.
Thus, the original problem simplifies to \(x^{21}\). Understanding this process is crucial as it lays the groundwork for dealing with more sophisticated algebraic expressions efficiently.

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

simplify each algebraic expression. $$ -(2 x-3 y-6) $$

In Exercises \(1-10\), factor out the greatest common factor. $$6 x^{4}-18 x^{3}+12 x^{2}$$

Simplify each exponential expression $$ \left(10 x^{2}\right)^{-3} $$

Simplify each exponential expression $$ \frac{20 x^{24}}{10 x^{6}} $$

Simplify each exponential expression $$ \left(\frac{3 x^{4}}{y}\right)^{-3} $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Discrete 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.