Problem 30 - Algebra 2 and Trigonometry

Question

If \(25^{b+1}=x\) and \(5^{b}=y,\) express \(x\) in terms of \(y\)

Step-by-Step Solution

Answer

\(x = 25y^2\)

Step 1: Rewrite the equation for x

Start with the equation given for \(x\), which is \(25^{b+1} = x\). Since \(25 = 5^2\), rewrite the equation as \((5^2)^{b+1} = x\).

Step 2: Apply the Power of a Power Property

Using the power of a power property \((a^m)^n = a^{m \cdot n}\), rewrite \((5^2)^{b+1}\) as \(5^{2(b+1)}\). So, we have \(x = 5^{2(b+1)}\).

Step 3: Simplify the Exponents

Distribute the 2 in the exponent: \(5^{2(b+1)} = 5^{2b + 2}\). Hence, the equation is \(x = 5^{2b + 2}\).

Step 4: Express x in Terms of y

We are given that \(5^b = y\). Rewrite it in terms of \(5^{2b}\) by squaring both sides: \((5^b)^2 = y^2\), thus \(5^{2b} = y^2\).

Step 5: Substitute for x

From Step 3, we have \(x = 5^{2b+2}\). Notice that \(5^{2b+2} = 5^{2b} \cdot 5^2\). Substitute \(5^{2b} = y^2\) from Step 4, so \(x = y^2 \cdot 25\).

Step 6: Final Expression for x

Thus, the expression for \(x\) in terms of \(y\) is \(x = 25y^2\).

Key Concepts

Power of a Power Property

The Power of a Power Property is a fundamental rule in mathematics that helps simplify expressions involving exponents. It states that when you raise a power to another power, you multiply the exponents. For example,

• If you have \[ (a^m)^n \], you can rewrite it as \[ a^{m \cdot n} \].

This property is particularly useful while dealing with exponential equations or expressions, like in the exercise we just saw. In our case, the expression \[ (5^2)^{b+1} \] was simplified by using the power of a power property. Instead of continuously calculating large values, we used this property to simplify \[ 25^{b+1} \] to \[ 5^{2(b+1)} \]. This makes further calculations much easier to handle. It’s a crucial step that sets the groundwork for simplifying even more complex exponentials.

Simplifying Exponents

Simplifying exponents means to rewrite exponential expressions in their simplest form, making calculations easier. This involves breaking down and reducing powers logically.

• For example, in our exercise, we transformed \[ 5^{2(b+1)} \] to \[ 5^{2b + 2} \].
• This process involves applying basic algebra to expand and combine like terms, effectively managing larger numbers by working with their exponents first.

This step is crucial in solving exponential equations, as it converts complex expressions into more manageable pieces by focusing on the exponents, allowing us to see relationships clearly, such as pairing like terms. Simplification is a key mathematical skill that aids in clear problem-solving, and makes further steps less daunting.

Expressing Variables in Terms of Others

Expressing variables in terms of others is a useful algebraic process that involves using equations to relate different variables. In our exercise, we needed to express \( x \) in terms of \( y \). Often, this requires substituting known expressions or values into another equation.

• We started with \( x = 5^{2b+2} \) from our simplification process, and used the relationship \( 5^b = y \), provided in the exercise.
• By squaring both sides of \( 5^b = y \), we found \( 5^{2b} = y^2 \).
• This allowed us to substitute \( 5^{2b} = y^2 \) into our expression for \( x \), resulting in \(x = 5^{2b} \cdot 5^2 = y^2 \cdot 25 \).

This final step helped us express \( x \)in fully simplified terms of \( y \), creating a straightforward mathematical connection between the variables without additional complex calculations. This process highlights the power of algebraic manipulation in problem-solving by deriving meaningful relationships between variables.