Problem 12 - Precalculus Functions and Graphs

Question

Determine the number that can be used in place of the question mark to make the equation true. $$3^{?}=1$$

Step-by-Step Solution

Answer

The number is 0.

Step 1: Understand the equation

The equation given is \(3^{?} = 1\). The goal is to determine the exponent that makes this equation true.

Step 2: Recall properties of exponents

Any non-zero number raised to the power of 0 is equal to 1. Mathematically, this is represented as \(a^0 = 1\), where \(a eq 0\).

Step 3: Apply the property to the given equation

Since \(3\) is a non-zero number, and we want \(3^?\) to equal 1, we can use the property that \(3^0 = 1\).

Step 4: Conclusion

Therefore, the value that can replace the question mark to make the equation true is 0.

Key Concepts

Properties of Exponents

Exponents are powerful tools in mathematics that help to simplify large numbers and solve equations. They represent repeated multiplication. For example, \(3^4\) means \(3 \times 3 \times 3 \times 3\).

Here are the key properties of exponents:

• Product of Powers: When you multiply two exponents with the same base, you add their exponents: \(a^m \times a^n = a^{m+n}\).
• Quotient of Powers: When you divide two exponents with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
• Power of a Power: Raising an exponent to another exponent means you multiply the exponents: \((a^m)^n = a^{mn}\).
• Zero Exponent Property: Any non-zero number raised to the power of 0 equals 1: \(a^0 = 1\).

Remembering these properties can make working with exponents much easier. They help to simplify calculations and solve complex equations accurately.

Zero Exponent Rule

The Zero Exponent Rule is a special property of exponents. It states that any non-zero number raised to the power of 0 equals 1. Mathematically, this is written as \(a^0 = 1\), where \(a eq 0\).

Why does this work? Consider the pattern when decreasing exponents:

• \(3^3 = 27\)
• \(3^2 = 9\)
• \(3^1 = 3\)
• \(3^0 = 1\)

Each time the exponent decreases by 1, we divide the previous result by the base (in this case, 3). So, naturally, \(3^0 = 1\) maintains the pattern.

This rule is key to solving equations where the answer must equal 1, just like in the exercise \(3^? = 1\). The only exponent that makes any non-zero number equal to 1 is the zero exponent.

Exponential Equations

Exponential equations involve variables in their exponents. They can seem tricky at first, but understanding the properties of exponents and the zero exponent rule can make them easier to solve.

Let's look at how to solve exponential equations:

• Equal Bases: If the bases on both sides of the equation are the same, set the exponents equal to each other. For example, to solve \(2^{x+1} = 2^3\), we set \(x+1 = 3\).
• Solving Different Bases: If the bases are different, try to rewrite them with the same base. For example, \(4^x = 2^3\) can be rewritten as \((2^2)^x = 2^3\), which simplifies to \(2^{2x} = 2^3\).
• Using Logarithms: If rewriting isn't possible, use logarithms. They help to bring down the exponent and solve for the variable.

Recognizing patterns and applying these rules makes solving exponential equations more manageable. Always start by simplifying the equation with exponent properties.