/*! 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 2 Simplify the expression complete... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 2

Simplify the expression completely. $$10^{\log (A B)}$$

Short Answer

Expert verified
The expression simplifies to \( AB \).

Step by step solution

01

Review the Problem

We need to simplify the expression \( 10^{\log (A B)} \).
02

Recall the Power Rule for Logarithms

The power rule for logarithms states that \( a^{\log_a(b)} = b \) where \( a > 0 \) and \( a eq 1 \). In this case, the base \( a \) is 10.
03

Apply the Power Rule

By applying the power rule, since our expression matches the form, \( 10^{\log_{10}(A B)} = A B \).
04

Simplify Final Expression

The simplified form of the expression \( 10^{\log (A B)} \) is just \( A B \).

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.

Exponentiation
Exponentiation is a key mathematical operation involving numbers raised to a power, also known as the base and the exponent. In the expression \( 10^{\log (A B)} \), 10 is the base, and \( \log (A B) \) is the exponent. This operation means the base is multiplied by itself a certain number of times, determined by the exponent. However, when the exponent involves logarithms, it often signifies a different form of relationship.To understand the role of exponentiation better:
  • The base (10, in this case) indicates the number repeatedly multiplied.
  • The exponent can be any real number, and when it's a logarithm, it represents the power to which the base must be raised to obtain another number.
  • Special rules, such as the power rule for logarithms, help simplify these expressions efficiently.
By understanding how the base and exponent interact, particularly with logarithms, students can simplify complex expressions effectively.
Simplification Techniques
Simplification techniques are essential in making complex mathematical expressions more manageable. In the problem \( 10^{\log (A B)} \), simplification involves applying specific rules and understanding properties, like the power rule for logarithms.Simplification begins by acknowledging:
  • The power rule: For any base \( a > 0 \), \( a^{\log_a(b)} = b \). This rule helps reduce the expression to a simpler form.
  • Using the power rule in this case simplifies the expression from \( 10^{\log (A B)} \) to \( A B \). This transformation happens because the base of the exponent (10) matches the logarithm's base, making the expression fit the power rule perfectly.
  • Recognizing patterns and applying the right rules can make complicated expressions straightforward and easier to work with.
This approach underscores how powerful simplification techniques can be when handling complex mathematical tasks, highlighting the importance of understanding and utilizing appropriate rules.
Mathematical Notation
Mathematical notation provides a standardized way to represent mathematical concepts and relationships succinctly. In the expression \( 10^{\log (A B)} \), notation is not just for representation but also for revealing underlying mathematical relationships.Key insights about mathematical notation:
  • Exponentiation is denoted by a base and an exponent, clearly shown with \( 10^{\log (A B)} \).
  • Logarithms are represented as \( \log \,\), indicating the inverse operation of exponentiation, and they describe the power needed to raise the base to a specific number.
  • Proper understanding and use of notation aids in communicating and solving mathematical problems effectively.
By learning how to read and interpret mathematical notation, students gain deeper insights into solving expressions and enjoying the elegance of mathematics. Well-applied notation can significantly ease the process of problem-solving and help in the accurate communication of complex ideas.

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

Is the statement true or false? Assume that \(\lim _{x \rightarrow 7^{+}} f(x)=2\) and \(\lim _{x \rightarrow 7^{-}} f(x)=-2 .\) Explain. If \(f(7)=2,\) the function \((f(x))^{2}\) is continuous at \(x=7\).

A baseball hit at an angle of \(\theta\) to the horizontal with initial velocity \(v_{0}\) has horizontal range, \(R,\) given by $$ R=\frac{v_{0}^{2}}{g} \sin (2 \theta) $$ Here \(g\) is the acceleration due to gravity. Sketch \(R\) as a function of \(\theta\) for \(0 \leq \theta \leq \pi / 2 .\) What angle gives the maximum range? What is the maximum range?

This problem introduces the arccosine function, or inverse cosine, denoted by \(\cos ^{-1}\) on most calculators. (a) Using a calculator set in radians, make a table of values, to two decimal places, of \(g(x)=\arccos x\) for \(x=-1,-0.8,-0.6, \ldots, 0, \ldots, 0.6,0.8,1\) (b) Sketch the graph of \(g(x)=\arccos x\) (c) Why is the domain of the arccosine the same as the domain of the arcsine? (d) What is the range of the arccosine? (e) Why is the range of the arccosine not the same as the range of the arcsine?

The Bay of Fundy in Canada has the largest tides in the world. The difference between low and high water levels is 15 meters (nearly 50 feet). At a particular point the depth of the water, \(y\) meters, is given as a function of time, \(t\), in hours since midnight by $$y=D+A \cos (B(t-C))$$ (a) What is the physical meaning of \(D ?\) (b) What is the value of \(A ?\) (c) What is the value of \(B ?\) Assume the time between successive high tides is 12.4 hours. (d) What is the physical meaning of \(C ?\)

Are the statements true or false? Give an explanation for your answer. The family of functions \(y=a \sin x,\) with \(a\) a positive constant, all have the same period.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Statistics

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.