Solve. ((1)/(8))^x=16^1-x

Name: Problem 35, Chapter 12: Algebra A Combined Function
Uploaded: 2022-02-11T13:12:31-08:00
Duration: 1 min 30 s
Channel: Gregory Higby
Description: Solve. ((1)/(8))^x=16^1-x

step 1 The whole premise on this problem is to get the same base between 1 8th and 16. And it might not

Question

Solve. $$ \left(\frac{1}{8}\right)^{x}=16^{1-x} $$

   Solve.
$$
\left(\frac{1}{8}\right)^{x}=16^{1-x}
$$

Algebra A Combined Function

Elayn Martin-Gay 4th Edition

Chapter 12, Problem 35

Instant Answer

Solved by Expert Gregory Higby

02/11/2022

Step 1

We know that $\frac{1}{8}$ is $2^{-3}$ and $16$ is $2^4$. So, the equation becomes: $$ \left(2^{-3}\right)^{x}= \left(2^{4}\right)^{1-x} $$ Show more…

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Solve. $$ \left(\frac{1}{8}\right)^{x}=16^{1-x} $$

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Key Concepts

Exponential Equations

This concept involves equations where the variable appears in the exponent. The goal is to manipulate the equation so that the exponents can be equated, often by rewriting the bases on both sides or by applying logarithms to simplify the equation.

Common Base Method

This method consists of expressing both sides of an exponential equation in terms of the same base. When possible, it allows for the direct comparison of the exponents, turning a complicated exponential equation into a simpler linear or algebraic equation.

Properties of Exponents

These properties include rules such as a^(b*c) = (a^b)^c, the negative exponent rule (a^(-n) = 1/a^n), and the product rule (a^b * a^c = a^(b+c)). Understanding these rules is crucial for rewriting expressions and solving exponential equations by manipulating and simplifying the terms involved.

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