Problem 30 - Algebra A Combined Function

Question

By inspection, find the value for \(x\) that makes each statement true. \(4^{x}=1\)

Step-by-Step Solution

Answer

The value of \(x\) is 0.

Step 1: Understand the Equation

The given equation is \(4^{x} = 1\). Our task is to find the value of \(x\) that makes this equation true.

Step 2: Recognize the Power Property

Recognize that any number raised to the power of zero is 1. Mathematically, \(a^0 = 1\) for any non-zero number \(a\). In this case, \(4^0 = 1\).

Step 3: Apply the Property to Solve for x

Using the property from Step 2, we know that if \(4^x = 1\), then \(x\) must be 0. Therefore, \(x = 0\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers: the base and the exponent. It is often written in the form of \(a^b\). Here, \(a\) represents the base and is the number being multiplied by itself. The number \(b\), or the exponent, indicates how many times the base is multiplied.
For example, if you have \(2^3\), this means 2 is used as a factor three times:

• \(2 \times 2 \times 2 = 8\)

Exponentiation is a fundamental concept in mathematics as it offers a concise way to represent repeated multiplication. You'll frequently encounter this operation across various math levels and in different contexts, such as when calculating areas, volumes, and even in scientific notation.

• It's essential to understand that exponentiation is not the same as simple multiplication.
• It is an operation that efficiently expresses large numbers and calculations that would otherwise be cumbersome.

Powers of Numbers

A power of a number refers to the result obtained when a number is raised to an exponent. It shows us how many times to use the number in a multiplication. This is what we see in expressions like \(3^2\) ("three squared") and \(5^4\) ("five to the fourth power").

• For \(3^2\): it is \(3 \times 3 = 9\).
• For \(5^4\): it is \(5 \times 5 \times 5 \times 5 = 625\).

Understanding powers of numbers is crucial because they simplify expressions and calculations that involve repetition. Powers also help in understanding relationships in equations and in solving scientific problems where growth patterns emerge, such as population growth or radioactive decay.
A special case arises when the power is zero, leading us to the next important concept about exponents.

Zero Exponent Property

The zero exponent property is a unique and highly useful rule in mathematics. It states that any non-zero number raised to the power of zero is always equal to 1. So, no matter what base you start with, if the exponent is 0, the result will always be 1. This can be expressed as:\[a^0 = 1 \] for any non-zero number \(a\).

• For example: \(7^0 = 1\), \(1000^0 = 1\), even \((-3)^0 = 1\).

This rule simplifies expressions and makes calculations easier by eliminating the need to perform multiplication when the exponent is zero. This property also gives us a solution to expressions like \(4^x = 1\), where by applying the zero exponent property, we quickly see that \(x\) must be 0, since \(4^0 = 1\). Understanding this property helps build a strong foundation for tackling more complex algebraic manipulations efficiently.