Problem 13 - Precalculus Functions and Graphs

Question

Determine the number that can be used in place of the question mark to make the equation true. $$16^{?}=2$$

Step-by-Step Solution

Answer

? = \frac{1}{4}

Step 1: Understand the Problem

The equation given is \(16^{?}=2\). We need to determine what power of 16 equals 2.

Step 2: Express 16 as a Power of 2

Recognize that 16 can be expressed as a power of 2. Since \(16 = 2^4\), we can rewrite the equation: \((2^4)^{?} = 2\).

Step 3: Simplify the Equation

Use the property of exponents that states \((a^m)^n = a^{m \times n}\). This simplifies to: \(2^{4?} = 2\).

Step 4: Set the Exponents Equal

Since the bases are the same, set the exponents equal: \(4? = 1\).

Step 5: Solve for the Unknown

Solve for the unknown by dividing both sides by 4: \(? = \frac{1}{4}\).

Key Concepts

Properties of Exponents

Understanding the properties of exponents is crucial for solving equations like the one given in the exercise. Exponents allow us to express repeated multiplication in a compact form. Here are some key properties to remember:

1. **Product of Powers**: When multiplying two exponents with the same base, you add the exponents. For example, \[a^m \times a^n = a^{m+n}\].
2. **Quotient of Powers**: When dividing two exponents with the same base, you subtract the exponents. For example, \[\frac{a^m}{a^n} = a^{m-n}\].
3. **Power of a Power**: When raising an exponent to another power, you multiply the exponents. For example, \[(a^m)^n = a^{m\times n}\].

In the given exercise, we used the property \[(a^m)^n = a^{m\times n}\] to simplify \[(2^4)^? = 2\] to \[2^{4?} = 2\]. This property allowed us to work directly with the exponents and solve for the unknown.

Solving Exponential Equations

Solving exponential equations involves matching the bases and then equating the exponents. Here are steps to solve exponential equations:

1. **Express All Terms Using the Same Base**: Rewrite all terms so they have the same base. In our exercise, we rewrote 16 as \[2^4\].
2. **Use Properties of Exponents**: Apply the properties of exponents to simplify the equation. Once simplified, the equation should have the same base on both sides. For our exercise, this gave us \[2^{4?} = 2\].
3. **Set the Exponents Equal**: Since the bases are the same, you can set the exponents equal to each other. For example, \[4? = 1\].
4. **Solve for the Unknown**: Solve the resulting equation for the unknown variable. In the exercise, we solved \[4? = 1\] by dividing both sides by 4, which gave \(? = \frac{1}{4}\).

Following these steps systematically helps in solving any exponential equations with confidence.

Expressing Numbers as Powers

A key part of working with exponents is being able to express numbers as powers of other numbers. For instance, recognizing that 16 is \[2^4\] can simplify many problems.

Here are some tips:

• **Learn Common Powers**: Memorize common bases and their powers. For instance, \[2^2 = 4, 2^3 = 8, 2^4 = 16\].
• **Prime Factorization**: Break down numbers into prime factors. For 16, you get \[16 = 2 \times 2 \times 2 \times 2 = 2^4\].

In the given exercise, identifying that 16 is \[2^4\] was crucial for converting the original equation into a form that allowed us to solve for the unknown exponent. Doing this transformation simplifies the problem by allowing you to work with a common base, making other properties of exponents applicable and the equations solvable.