Problem 95 - Intermediate Algebra

Question

Look Alikes \(\cdots\) 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: Recognize the problem type

We are given equations with exponential expressions and need to solve for the variable in the exponent. This is a typical exponential equation problem.

Step 2: Analyze the base of exponents

For both sub-problems (a) and (b), the base of the exponent on the left side is \(\frac{2}{3}\). We must express the right-hand side with the same base raised to some power.

Step 3: Simplify the right side of equation (a)

Identify \(\frac{8}{27}\) as being \(\left(\frac{2}{3}\right)^3\), since both 8 and 27 are powers of 2 and 3 respectively: \(8 = 2^3\) and \(27 = 3^3\). Therefore, \(\frac{8}{27} = \left(\frac{2}{3}\right)^3\).

Step 4: Set the exponents equal in (a)

Since \(\left(\frac{2}{3}\right)^{6-x} = \left(\frac{2}{3}\right)^3\), the exponents must be equal: \(6-x = 3\).

Step 5: Solve for x in (a)

Solve the equation \(6-x = 3\) by isolating \(x\): \(x = 6 - 3 = 3\).

Step 6: Simplify the right side of equation (b)

Identify \(\frac{16}{81}\) as being \(\left(\frac{2}{3}\right)^4\), since both 16 and 81 are powers of 2 and 3 respectively: \(16 = 2^4\) and \(81 = 3^4\). Therefore, \(\frac{16}{81} = \left(\frac{2}{3}\right)^4\).

Step 7: Set the exponents equal in (b)

Since \(\left(\frac{2}{3}\right)^{6-x} = \left(\frac{2}{3}\right)^4\), the exponents must be equal: \(6-x = 4\).

Step 8: Solve for x in (b)

Solve the equation \(6-x = 4\) by isolating \(x\): \(x = 6 - 4 = 2\).

Key Concepts

Exponents

Exponents are a way of representing repeated multiplication of the same number. For instance, if we have a number like \(2^3\), it means that the number 2 is multiplied by itself three times: \(2 \times 2 \times 2 = 8\). In general, an expression such as \(a^b\) indicates that the base \(a\) is raised to the exponent \(b\). This concept is crucial in understanding exponential equations, where the variable is often found in the exponent position.
Exponents simplify the representation of very large or very small numbers by allowing us to use powers rather than long multiplication sequences. They are present in various mathematical applications, including growth processes and decay models. The key property of exponents used in solving equations is that if two exponential expressions with the same base are equal, then their exponents must also be equal, for instance, if \(b^x = b^y\), then \(x = y\).
Understanding exponents also involves learning about the different rules, such as product of powers, power of a power, and negative exponents, which are often used in algebra to manipulate and solve equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations such as addition, subtraction, multiplication, and division. An algebraic expression can be as simple as \(3x + 2\) or more complex like \((\frac{2}{3})^{6-x}\).
When dealing with exponents in algebraic expressions, it's important to know how to simplify, combine, or break down terms. For example, understanding how to convert \(\frac{8}{27}\) into \((\frac{2}{3})^3\) is a part of simplifying expressions to make them easier to solve. This often requires recognizing patterns and properties of numbers, such as identifying powers and roots.
Manipulating algebraic expressions is a fundamental skill in algebra that allows solving equations and inequality problems. By understanding how to manipulate these expressions, you learn how to move terms around, combine like terms, and simplify the expression to facilitate solving the equation.

Solving Equations

Solving equations often means finding the value of the variable that makes the equation true. In exponential equations, the variable is usually in the exponent, which can initially seem daunting.
To crack exponential equations, follow these steps:

• Identify the bases involved and see if they can be written as powers of the same number.
• Once expressions on both sides have the same base, equate the exponents.
• Solve the resulting simple algebraic equation to find the value of the variable.

For example, in the equation \((\frac{2}{3})^{6-x} = (\frac{2}{3})^3\), the bases are already the same. Thus, equating the exponents gives: \(6-x = 3\). Solving this gives \(x = 3\).
Understanding each step in solving exponential equations ensures a solid grasp of how mathematical principles apply, providing clarity and making problem-solving a more straightforward process. With practice, these techniques become second nature, and solving such equations becomes a seamless process.