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Chapter 4: Problem 14

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

Short Answer

Expert verified
The value of the exponent is 1.

Step by step solution

01

Understand the Equation

The given equation is \(16^{?}=16\). This means we need to determine the value of the exponent that makes the base 16 equal to 16.
02

Rewrite the Base Term

Rewrite the number 16 in exponential form. We know that \(16 = 16^1\).
03

Equate the Exponents

Since both sides of the equation have the same base (16), we can set the exponents equal to each other: \( ? = 1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Base and exponent
To understand exponential equations, we first need to understand the terms 'base' and 'exponent'. An exponential expression takes the form of \(a^b\), where \(a\) is the base and \(b\) is the exponent.
The base in an exponential expression is the number that gets multiplied. The exponent tells us how many times the base is multiplied by itself. For example, in \(2^3\), 2 is the base, and 3 is the exponent.
  • \(2^3 = 2 \times 2 \times 2 = 8\)
The base can be any real number, and the exponent can be positive, negative, or zero. Understanding this basic relationship is essential for solving exponential equations.
Exponential equations
An exponential equation is an equation in which the variables appear as exponents. An example is \(16^?=16\).
To solve such equations, we need to manipulate the equation until both sides have the same base. Once the bases are the same, we can equate the exponents.
Here is a simple example:
Suppose we have \(2^x = 8\). We know that 8 can be written as \(2^3\). Therefore, \(2^x = 2^3\). Since the base (2) is the same, we can equate the exponents:
  • \(x = 3\)
This method of rewriting the numbers to have a common base is vital for solving exponential equations.
Equating exponents
Equating exponents is the final step in solving exponential equations.
Once we have the same base on both sides of the equation, we can simply set the exponents equal to each other.
For the original problem, \(16^?=16\), we first recognize that 16 can be rewritten in exponential form as \(16^1\).
Therefore, our equation becomes \(16^? = 16^1\). Since the bases are the same (16), we can equate the exponents on both sides:
  • \(? = 1\)
This tells us that the exponent needed to make the equation true is 1.
This straightforward approach makes solving such problems more manageable once you grasp the importance of equating exponents.

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Most popular questions from this chapter

Noise Pollution The level of a sound in decibels (db) is determined by the formula $$\text { sound level }=10 \cdot \log \left(I \times 10^{12}\right) \mathrm{db}$$ where \(I\) is the intensity of the sound in watts per square meter. To combat noise pollution, a city has an ordinance prohibiting sounds above 90 db on a city street. What value of \(I\) gives a sound of 90 db?

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