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Chapter 5: Problem 2

For an integer \(n\) greater than 1 , if \(b^n=a\), then \(b\) is a(n) of \(a\).

Short Answer

Expert verified
The base \(b\) can be declared the root of \(a\).

Step by step solution

01

Understanding the concepts

Remember the basic rules of exponents: a base (\(b\)) is raised to a power (\(n\)), the result is given (\(a\)). In this exercise, the exponentiation operation is being described.
02

Identify the missing term

The base (\(b\)) is raised to a power (\(n\)) equals \(a\), given this, we've got to find out a name of \(b\) with respect to \(a\).
03

Analyze and Conclude

The base of an exponential expression, in relation to the result, is referred to as a 'root'. So the base \(b\) can be considered a root of \(a\).

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Key Concepts

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

Integer Exponents
In mathematics, **integer exponents** refer to the power to which a base number is raised when the exponent is an integer (a whole number, positive or negative, including zero).
When you see something like \(b^n\), \(b\) is the base and \(n\) is the exponent. Suppose \(n\) is an integer:
  • **Positive integers**: If \(n\) is positive, say \(n = 3\), it means multiply the base \(b\) by itself three times, i.e., \(b^3 = b \times b \times b\).
  • **Zero as an exponent**: When \(n = 0\), \(b^0 = 1\) for any non-zero \(b\). This is because zero exponents always yield 1.
  • **Negative integers**: When \(n\) is negative, for example \(n = -2\), we take the reciprocal of the base raised to the positive power, i.e., \(b^{-2} = \frac{1}{b^2}\).
These rules help manage and simplify exponential expressions and are crucial for solving equations involving exponents.
Roots
Conceptually, roots are closely tied to exponents. If an exponent determines how many times a base is multiplied by itself, a **root** performs the inverse operation.
Consider the expression \(b^n = a\). Here, \(b\) is regarded as the \(n\)-th root of \(a\). This means if you raise \(b\) to the power of \(n\), you should get \(a\).
  • **Square Root**: If \(n = 2\), then \(b\) is the square root of \(a\). So, \(b^2 = a\) implies \(\sqrt{a} = b\).
  • **Cube Root**: If \(n = 3\), \(b\) is considered the cube root of \(a\), written as \(\sqrt[3]{a} = b\).
The root helps in understanding the base when an exponential expression is involved. Knowing this makes it easier to identify and solve equations where you need to "reverse" exponentiation.
Exponential Expressions
An **exponential expression** is any expression that involves an exponent. It follows a basic form: \(b^n\), where \(b\) is the base and \(n\) is the exponent. These expressions are powerful in representing very large or very small numbers compactly.
In any exponential expression, each part plays a crucial role:
  • **Base (\(b\))**: This is the number that gets multiplied by itself. It determines the size affected by the exponent.
  • **Exponent (\(n\))**: It shows how many times the base multiplies itself. Higher the exponent, the larger the exponential growth or decay.
Applications of exponential expressions are everywhere, from calculating compound interest to understanding geometric growth. It's important to grasp the relationship between the base and exponent, especially in expressions like \(b^n = a\), to tackle complex problems by simplifying or transforming them into more manageable forms, often involving roots.

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Most popular questions from this chapter

\(f(x)=\sqrt[3]{x^2+10 x}, g(x)=\frac{1}{4} f(-x)+6\)

\(x^2=100-y^2\)

\(x^2+y^2=9\)

The maximum hull speed \(v\) (in knots) of a boat with a displacement hull can be approximated by \(v=1.34 \sqrt{\ell}\), where \(\ell\) is the waterline length (in feet) of the boat. Find the inverse function. What waterline length is needed to achieve a maximum speed of \(7.5\) knots?

\(f(x)=\sqrt{x}, g(x)=2 \sqrt{x-1}\)
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