Question

If \(9^{x}=25,\) what does \(3^{x}\) equal?

Short answer

\(3^x = 5\).

Step-by-step solution

- Rewrite the Base

First, express the base 9 in terms of base 3. Note that 9 can be written as \(3^2\). So, rewrite the given equation: \(9^x = 25\) becomes \((3^2)^x = 25\).

- Simplify the Exponent

Next, use the power rule of exponents, which states that \((a^m)^n = a^{mn}\). This means that \((3^2)^x\) is the same as \(3^{2x}\). Thus, the equation now is:\(3^{2x} = 25\).

- Solve for \(3^x\)

We need to find the value of \(3^x\). Notice that the current equation \(3^{2x} = 25\) can be rewritten by taking the square root of both sides. Taking the square root yields:\(3^x = \sqrt{25}\).Since \sqrt{25} = 5,\(3^x = 5\).

Key concepts

Exponentiation

Exponentiation is a fundamental math operation that involves raising a number, known as the base, to the power of an exponent. This operation is written as \( a^n \), where \( a \) is the base and \( n \) is the exponent. The result represents multiplying the base by itself \( n \) times, such as \( 3^4 = 3 * 3 * 3 * 3 = 81 \). It's a key concept in algebra and higher-level mathematics.

• The base is the number you're multiplying.
• The exponent tells you how many times to multiply the base by itself.

Exponentiation comes with several rules:
1. Any number raised to the power of zero is 1, written as \( a^0 = 1 \).
2. Any number raised to the power of one is itself, written as \( a^1 = a \).
3. When raising a product to a power, apply the power to each factor: \( (ab)^n = a^n b^n \).Understanding these rules helps solve more complex problems, such as the one in our exercise.

Base Conversion

Base conversion involves changing numbers from one base to another to simplify calculations or to find an equivalent expression. In our exercise, we converted base 9 into base 3. This helps relate the problem to more familiar or manageable numbers. Let's look at the details:

• \texample: we know that \( 9 = 3^2 \), so instead of 9, we use \( 3^2 \).
• \tThis transforms the equation \( 9^x = 25 \) into \( (3^2)^x = 25 \).

This simplification is crucial as it allows us to use the same base for easy comparison and calculations. Working within a consistent base facilitates the application of exponent rules and makes solving the equation straightforward.

Power Rule

The power rule is essential for manipulating expressions involving exponents. It states that \( (a^m)^n = a^{mn} \). This rule simplifies exponents in expressions involving multiple layers of exponentiation. In our exercise, we applied this rule to simplify the rewritten equation:
1. Start with \( (3^2)^x \).
2. Apply the power rule: \( (3^2)^x = 3^{2x} \).
This brings us to a simpler form, allowing us to easily compare the two sides of the equation. Applying the rule appropriately aids in steps like solving for \( 3^x \) by eventually reaching an equation that's easier to handle or recognize.