/*! 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 97 Solve each equation. $$5^{x^{2... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 97

Solve each equation. $$5^{x^{2}-12}=25^{2 x}$$

Short Answer

Expert verified
The solutions to the equation are \(x = 6\) and \(x = -2\).

Step by step solution

01

Express both sides with the same base

Rewrite the equation such that both sides have the same base. Since 25 is a power of 5, rewrite \(25 = 5^{2}\). So, the equation becomes \(5^{x^{2}-12} = (5^{2})^{2x}\). Now simplify the right side using the property of exponentiation \(a^{m \cdot n} = a^{m \cdot n}\), to obtain \(5^{x^{2}-12} = 5^{4x}\).
02

Make the exponents equal

Since now both sides of the equation share the same base, equate their exponents. This gives you a simple algebraic equation: \(x^{2}-12 = 4x\).
03

Re-arrange and solve quadratic equation

Re-arrange the equation from step 2 to have it in the standard quadratic equation format. We get: \(x^{2} - 4x -12 = 0\). Now factorize and solve for \(x\): \((x - 6)(x + 2) = 0\). Solving this gives two possible solutions \(x = 6\) and \(x = -2\).

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!

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(x=\frac{1}{k} \ln y,\) then \(y=e^{k x}\)

The formula \(A=36.1 e^{0.0126 t}\) models the population of California, \(A,\) in millions, \(t\) years after 2005 a. What was the population of California in \(2005 ?\) b. When will the population of California reach 40 million?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(\log (x+3)=2,\) then \(e^{2}=x+3\)

Because the equations $$\log (3 x+1)=5 \text { and } \log (3 x+1)=\log 5$$ are similar, I solved them using the same method.

Solve: $$\sqrt{2 x-1}-\sqrt{x-1}=1$$
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Decision 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.