Problem 95 - Elementary and Intermediate Algebra

Question

a. \(\left(\frac{2}{3}\right)^{6-x}=\frac{8}{27}\) b. \(\left(\frac{2}{3}\right)^{6-x}=\frac{16}{81}\)

Step-by-Step Solution

Answer

a. \(x = 3\); b. \(x = 2\)

Step 1: Express Right Side in Powers of Base

First, express the right side of the equation as a power of \(\frac{2}{3}\).For part a: \(\frac{8}{27}\) can be written as \(\left(\frac{2}{3}\right)^3\), since \((2^3 = 8)\) and \((3^3 = 27)\).For part b: \(\frac{16}{81}\) can be written as \(\left(\frac{2}{3}\right)^4\), since \((2^4 = 16)\) and \((3^4 = 81)\).

Step 2: Set Exponents Equal

Since the bases are the same, set the exponents equal to each other.For part a: \[ 6 - x = 3 \]For part b: \[ 6 - x = 4 \]

Step 3: Solve for \(x\)

Rearrange the equation to solve for \(x\).For part a: \[ x = 6 - 3 = 3 \]For part b:\[ x = 6 - 4 = 2 \]

Key Concepts

Understanding Exponents

Exponents are a way of expressing repeated multiplication of the same number. They form the backbone of many aspects of algebra and are often encountered when solving exponential equations.
When you have a number termed as the base raised to an exponent (written as \(a^n\)), the exponent tells you how many times to multiply the base \(a\) by itself. For example, \(3^2\) means \(3 \times 3\), which results in 9.
Exponents allow for concise expression of very large numbers. This powers notation can represent real-world quantities like population growth or radioactive decay efficiently.

• When multiplying terms with the same base, simply add the exponents: \(a^m \times a^n = a^{m+n}\).
• When dividing terms with the same base, subtract the exponents: \(a^m / a^n = a^{m-n}\).

Understanding how exponents work will help you solve various problems where such notation is used.

Steps to Solving Exponential Equations

Solving exponential equations, where the unknown is in the exponent, involves specific strategies. These include matching the bases and setting exponents equal when the bases are similar.

Let's consider an equation like \(\left(\frac{2}{3}\right)^{6-x} = \frac{8}{27}\):

• Step 1: Express the Problem Logically
  First, convert \(\frac{8}{27}\) to a power with \(\frac{2}{3}\) as the base. Recognize this fraction is \(\left(\frac{2}{3}\right)^3\).


• Step 2: Equalize the Exponents
  Once both sides are expressed with the same base, you equate the exponents. Here, \(6-x=3\) because the bases are now equivalent.


• Step 3: Solve for the Variable
  Solve the resulting equation for \(x\). For our example, \(x = 6 - 3 = 3\).

By practicing these steps, you can tackle various exponential equations confidently, enhancing your algebraic skills.

Applying the Power Rules

Power rules are essential when dealing with exponents, especially in complex equations. These rules help you manipulate expressions effectively.

Two primary power rules are essential:

• Product of Powers Rule
  When you multiply like bases, add the exponents: \(a^m \times a^n = a^{m+n}\). This rule helps simplify expressions where terms with the same base are multiplied together.


• Quotient of Powers Rule
  When dividing like bases, subtract the exponents: \(a^m / a^n = a^{m-n}\). This is helpful when simplifying fractions that share the same base.

These rules are very useful for streamlining the solution process. For example, if you have a complex term such as \(\frac{a^5}{a^2}\), you can quickly determine it's equal to \(a^{3}\) using the quotient of powers rule.
With practice, applying power rules can become second nature, making complex exponential problems much more manageable.