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Chapter 4: Problem 28

Identify the base and the exponent in each expression. $$4^{x}$$

Short Answer

Expert verified
In the expression \(4^{x}\), the base is 4 and the exponent is \(x\).

Step by step solution

01

Understand the exponential notation

An exponential expression is typically written in the form \(b^{p}\), where \(b\) is called the base and \(p\) is the exponent or power.
02

Identify the base

In the expression \(4^{x}\), the base is the part of the expression that is being raised to a power. Here that is the number 4, so the base of this expression is 4.
03

Identify the exponent

In the expression \(4^{x}\), the exponent is the part that signifies how many times the base is multiplied by itself. Here that is \(x\), so the exponent of this exponential expression is \(x\).

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Key Concepts

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

Understanding the Base
In an exponential expression, the base is a crucial component. It indicates the number being multiplied repeatedly. For example, in the expression \(4^{x}\), the base is the number 4. This is the fixed part of the expression that establishes what value is being raised to a certain power.
  • Think of the base as the "actor" in a repeated multiplication story.
  • The base is the value that will be exponentially expanded, like a starting point.
This helps simplify how we view expressions, no matter how complex or large the exponents become, as the base easier to spot and understand.
Understanding the Exponent
The exponent in an exponential expression specifies how many times the base is multiplied by itself. It serves as an indicator of repetition. In the expression \(4^{x}\), \(x\) dictates the number of times 4 will multiply by 4 again, repeatedly.
  • Consider the exponent like a guidebook for multiplication.
  • It tells the base how many chapters to play out in the long multiplication tale.
With this understanding, even when variables or more complex numbers are exponents, you'll easily decipher what the expression means. The exponent is a small number written to the upper right of the base, but its impact is substantial.
Exploring Exponential Notation
Exponential notation is a short-hand method for writing repeated multiplication. Instead of writing 4 multiplied by itself \(x\) times as \(4 \times 4 \times \ldots\) up to \(x\) times, exponential notation condenses this to \(4^{x}\). This not only simplifies writing but also simplifies computation and understanding.
  • Saves time and space by condensing long multiplication.
  • Makes it easier to work with larger numbers and understand mathematical concepts.
Exponential notation is widely used in sciences and mathematics to express large numbers more conveniently. It's a powerful tool that allows for more effective communication of mathematical ideas.

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Most popular questions from this chapter

Find the difference when \(\left(-3 z^{3}-4 z+7\right)\) is subtracted from the sum of \(\left(2 z^{2}+3 z-7\right)\) and \(\left(-4 z^{3}-2 z-3\right)\)

Simplify or solve as appropriate. $$3 y(y+2)=3(y+1)(y-1)$$

Perform the operations. $$\begin{aligned}\\\ &-3 x^{3} y^{6}+2\left(x y^{2}\right)^{3}-(3 x)^{3} y^{6} \end{aligned}$$

Describe the steps involved in finding the product of a binomial and its conjugate.

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