Problem 28 - Intermediate Algebra

Question

For Problems \(1-34\), solve each equation. $$ 9^{x}=\frac{1}{27} $$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{3}{2}\).

Step 1: Recognize Base Relation

Notice that both sides of the equation can be expressed with the same base. Since 9 and 27 are powers of 3, we can rewrite the equation in terms of base 3. Recall that \(9 = 3^2\) and \(27 = 3^3\). Hence, the equation becomes \((3^2)^x = \frac{1}{3^3}\).

Step 2: Simplify the Equation

Apply the power of a power property \((a^m)^n = a^{mn}\) to the left side of the equation: \((3^2)^x\). This becomes \(3^{2x}\). Apply the negative exponent rule to the right side: \(\frac{1}{3^3} = 3^{-3}\). The equation is now \(3^{2x} = 3^{-3}\).

Step 3: Set Exponents Equal

Since the bases are the same, we equate the exponents to solve for \(x\). Therefore, we have the equation \(2x = -3\).

Step 4: Solve for x

To isolate \(x\), divide both sides of the equation by 2: \(x = \frac{-3}{2}\).

Key Concepts

Base Conversion

Understanding base conversion is essential when working with exponential equations, like the one in our exercise. Base conversion often allows us to express numbers in terms of a common base, making them easier to compare or manipulate.
To convert numbers into a common base, we identify a base that both numbers are powers of. In the given problem, both 9 and 27 can be expressed using base 3:

• 9 is the same as \( 3^2 \)
• 27 is the same as \( 3^3 \)

By expressing both sides of the equation in terms of base 3, we simplified the problem significantly. The equation \( 9^x = \frac{1}{27} \) becomes \( (3^2)^x = \frac{1}{3^3} \). This simplification is vital because it leads to a straightforward comparison of the exponents, avoiding unnecessary complications. Recognize, practice, and perfect base conversion whenever you encounter exponential equations.

Negative Exponents

Negative exponents represent reciprocals of numbers with positive exponents. This concept is crucial because it allows us to handle fractions in equations more easily.
When you see a negative exponent, it means take the reciprocal of the base raised to the corresponding positive exponent. For example, a negative exponent on the right side of our equation appears as \( \frac{1}{3^3} \), which can be rewritten as \( 3^{-3} \).
Understanding this concept can simplify many algebraic processes:

• If \( a^{-n} = \frac{1}{a^n} \), then rewriting fractions using negative exponents can help you compare or solve equations faster.
• They also allow us to maintain consistency when performing operations on both sides of an equation, as shown by aligning \( 3^{2x} \) on the left with \( 3^{-3} \) on the right.

Master the art of applying negative exponents, and you'll bring clarity to operations involving exponents and fractions.

Power of a Power

The power of a power rule is a fundamental property in exponential equations, which is used extensively in our solution. When you have an expression like \((a^m)^n\), it can be simplified to \(a^{mn}\).
In this problem, the left side of the equation, \((3^2)^x\), can be simplified using the power of a power rule. Applying the rule directly gives \(3^{2x}\).
This property simplifies expressions involving multiple layers of exponents by reducing them to a single exponent. Consider these benefits:

• It reduces the complexity of equations, making them easier to solve.
• This property is particularly useful when aligning exponents after converting bases.

The power of a power is not just a shortcut but a necessary tool in algebra that can streamline your mathematical work.