Problem 63 - Elementary and Intermediate Algebra

Question

Let \(f(x)=\left(\frac{1}{5}\right)^{x}\). Explain why we can rewrite the function equation as \(f(x)=5^{-x}\)

Step-by-Step Solution

Answer

\( f(x) = \left(\frac{1}{5}\right)^{x} \) is rewritten as \( f(x) = 5^{-x} \) because \( \frac{1}{5} \) is the same as \( 5^{-1} \).

Step 1: Understanding Exponentiation

The expression \(f(x) = \left(\frac{1}{5}\right)^{x}\) represents a function where \( \frac{1}{5} \) is the base and \( x \) is the exponent. In fraction form, \( \frac{1}{5} \) is the reciprocal of \( 5 \). Reciprocal means that \( \frac{1}{5} = 5^{-1} \).

Step 2: Rewriting the Base

Since the base \( \frac{1}{5} \) is the reciprocal of \( 5 \), we can express it as \( 5^{-1} \). Thus, the original function can be rewritten as \( f(x) = (5^{-1})^x \).

Step 3: Applying the Power of a Power Rule

When we have an expression like \( (a^m)^n \), it is equivalent to \( a^{m \cdot n} \). Applying this property to our expression \( (5^{-1})^x \), we multiply the exponents: \(-1 \cdot x\). This simplifies to \( 5^{-x} \).

Step 4: Final Expression

By applying the above steps, we can finally rewrite \( f(x) = \left(\frac{1}{5}\right)^{x} \) as \( f(x) = 5^{-x} \). This is the simplified form of the original function expression.

Key Concepts

Reciprocal

Reciprocals are all about flipping numbers. If you start with a number, like 5, the reciprocal is simply \( \frac{1}{5} \). Think of reciprocals as numbers that, when multiplied by the original number, give you 1. For instance, \( 5 \times \frac{1}{5} = 1\). They are particularly useful because they help in solving equations, especially when dealing with fractions involving exponents.In the context of functions, if you have a fraction like \( \frac{1}{5} \), it can be rewritten using exponents as \( 5^{-1} \). Just remember, when you see the negative exponent, it means reciprocal — it's a neat way to simplify expressions.

Power of a Power Rule

Exponentiation can often puzzle students, but the power of a power rule simplifies things. This rule states that when you have an exponent raised to another exponent, like in \( (a^m)^n \), it equals \( a^{m \cdot n} \). You basically multiply the exponents together. In the problem at hand, we apply this rule to \( (5^{-1})^x \). Here, \(-1\) is the exponent of 5, and \(x\) is another exponent applied to that result. By using the power of a power rule, we multiply the exponents: \(-1 \times x = -x\). This transforms our expression into \( 5^{-x} \). It's a powerful tool for manipulating exponential expressions and making them easier to interpret and solve.

Function Simplification

Simplification is about making expressions clearer or easier to solve while keeping the math accurate. When you encounter a complex function, like \( f(x) = \left(\frac{1}{5}\right)^{x} \), it might seem daunting at first. But by using mathematics principles, you can simplify this.First, recognize \( \frac{1}{5} \) as \( 5^{-1} \) (thanks to the reciprocal concept) to transform the base. Secondly, apply the power of a power rule to rewrite the expression into a more straightforward form, which in this case turns into \( 5^{-x} \). This is a cleaner and more concise version of the original function.Why simplify? It can reveal properties of the function more clearly, make calculations easier, and help us understand how the function behaves. It's a bit like organizing a messy desk — once everything is in order, you can find things much easier and work more effectively! So, whenever you simplify, think of it as tidying up the math world.