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Chapter 8: Problem 19

Fill in the blanks. If \(p^{n}=q, p\) is called an __________ of \(q\).

Short Answer

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Radix or Base

Step by step solution

01

Understanding Exponentiation

Exponentiation is a mathematical operation, written as \(p^n\), involving two numbers, the base \(p\) and the exponent \(n\). When \(p^n=q\), \(p\) is the base which is used to get the power of \(n\) equals \(q\). The base \(p\) is an important term here.
02

Identifying the Term

The term that is used for the base \(p\) in an exponentiation operation is 'Radix' or 'Base'.

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Key Concepts

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

Base in Exponentiation
The term "base" in mathematics is a fundamental concept in many operations. In the context of exponentiation, the base is the number that is multiplied by itself a certain number of times. In the expression \(p^n\), \(p\) is the base, which means it is the number being used as a factor repeatedly, and \(n\) is how many times the base is used as a factor. For example, in \(2^3\), the base is 2, which is multiplied by itself to get \(2 \times 2 \times 2 = 8\). This hierarchy is essential because it determines the value of the expression.
  • The base is always the number at the bottom of the expression in the exponential notation.
  • The base is pivotal in defining the resulting value of the power expression.
Understanding the role of the base helps clarify how complex powers are calculated and emphasizes its essential role in any exponential operation.
Radix: Another Name for Base
In mathematics, the word "radix" is synonymous with "base." While it is less commonly used than "base," knowing that radix refers to the base can be important for certain contexts, especially in number systems.
The term radix is often used when discussing numeral systems. For instance, in the decimal system, the radix is 10. It means every step in counting proceeds using powers of 10. Similarly, in binary systems used in computing, the radix is 2.
  • "Radix" helps in switching between various numeral systems.
  • It provides structure in representing numbers in different bases like binary, octal, and hexadecimal.
Being familiar with the term radix helps in understanding advanced mathematical contexts where different numeral systems are used.
Mathematical Operation in Exponentiation
Exponentiation is a mathematical operation that plays a significant role in algebra and higher mathematics. This operation involves two key elements: the base and the exponent. It's a way of representing repeated multiplication of the same number.
Unlike addition and multiplication which are binary operations involving two numbers, exponentiation is expressed with two parts but implies a repeated action. When you see \(p^n\), you can interpret it as multiplying the base \(p\) by itself \(n\) times.
  • This operation aims to express large numbers in a concise form.
  • It is a fundamental part of modern calculations, including those used in scientific and financial contexts.
Understanding exponentiation as a mathematical operation allows us to work more effectively with powers, roots, and logarithms, which are built upon this basic principle.

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Most popular questions from this chapter

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\sqrt{\frac{36 m^{2} n^{9}}{100 m n^{3}}}$$

Simplify each expression. Assume that all variables represent positive values. Assume no division by 0. SEE EXAMPLE 6. (OBJECTIVE 3) $$a^{3 / 5} a^{-1 / 2}$$

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\sqrt[3]{-81 x^{2} y^{3} z^{4}}$$

Simplify each expression. All variables of square root expressions represent positive numbers. Assume no division by 0. $$\sqrt{\frac{250}{49}}$$

Can you find a Pythagorean triple with \(b=10 ?\) Explain.
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