/*! 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 36 Solve each equation and check. ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 36

Solve each equation and check. \(5+7^{x}=6\)

Short Answer

Expert verified
The solution is \(x = 0\).

Step by step solution

01

Isolate the Exponential Term

First, subtract 5 from both sides of the equation to isolate the exponential term. This gives us:\[7^{x} = 6 - 5\]Simplifying the right side, we find:\[7^{x} = 1\]
02

Solve the Exponential Equation

Since the equation is now \(7^{x} = 1\), we need to find the value of \(x\) that makes this true. Recall that any non-zero number raised to the power of 0 is 1. Thus:\[7^{x} = 7^{0}\]This implies that:\[x = 0\]
03

Check the Solution

Substitute \(x = 0\) back into the original equation to verify:\[5 + 7^{0} = 6\]Since \(7^{0} = 1\), the left-hand side becomes:\[5 + 1 = 6\]This simplifies to \(6 = 6\), confirming our solution is correct.

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.

Isolating the Exponential Term
When faced with an exponential equation, the initial task is to isolate the exponential term, which is vital in simplifying the problem. For instance, in the equation we began with:
  • We had: \(5 + 7^x = 6\)
  • To isolate the exponential term \(7^x\), subtract 5 from both sides, leaving: \(7^x = 6 - 5\).
  • The equation simplifies to \(7^x = 1\).
Isolating the exponential term transforms the equation into a form that is ready to be solved, setting the stage for the next steps. It also simplifies the algebraic landscape, making it much easier to analyze and solve.
Exponential Equation
Once the exponential part of your equation is isolated, you encounter what is known as an exponential equation. In our context, we have the simplified equation:
  • \(7^x = 1\).
To solve this, recall the rule: any non-zero number raised to the power of zero is 1. This incredible property of exponents significantly aids in solving exponential equations:
  • Since \(7^0 = 1\), it implies \(x = 0\) right away.
Understanding this fundamental property of exponents is pivotal, as it allows us to find the solution almost immediately without further calculations. Exponential equations often hide solutions that are rooted in these simple exponent rules, making familiarity with these rules invaluable.
Checking Solutions
Even after solving an exponential equation, it's crucial to check your solution to ensure its accuracy. Substituting your result back into the original equation is a foolproof way to verify it. Here鈥檚 how we did it:
  • Substitute \(x = 0\) back into the original equation: \(5 + 7^0 = 6\).
  • Recognize that \(7^0 = 1\), so the equation becomes: \(5 + 1 = 6\).
  • This simplifies to \(6 = 6\), confirming that our solution is indeed correct.
This checking step serves as a confirmation that the solution not only satisfies the equation, but also reinforces the understanding of exponential properties. Always remember that the verification step is an essential component of solving equations, especially for catching potential errors early.

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

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ \frac{1}{5^{\frac{1}{2}}} $$

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ \left(\frac{3 m^{-3}}{2 n^{-2}}\right)^{-3} $$

In \(64-75,\) write each quotient as a product without a denominator. The variables are not equal to zero. $$ \frac{15 x^{-2} y^{2}}{3 x y^{5}} $$

In \(38-57,\) write each radical expression as a power with positive exponents and express the answer in simplest form. The variables are positive numbers. $$ \sqrt[4]{64 x^{5}} $$

In \(35-63,\) write each expression with only positive exponents and express the answer in simplest form. The variables are not equal to zero. $$ \frac{(2 a)^{-1}}{2(a)^{-2}} $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Calculus

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.