Problem 87 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(n\) is a natural number and \(a\) is a positive real number, then \(\left(a^{1 / n}\right)^{n}=a\)

Step-by-Step Solution

Answer

The statement is true. For any natural number \(n\) and positive real number \(a\), \(\left(a^{\frac{1}{n}}\right)^{n} = a\). This is because by the exponentiation rule, \(\left(a^{\frac{1}{n}}\right)^{n} = a^{\frac{1}{n} \times n} = a^1 = a\).

Step 1: Understanding the terms and given statement

We are given that n is a natural number and a is a positive real number. Natural numbers are the positive integers (1, 2, 3, …) and positive real numbers can be any positive number, including decimals and fractions. The given statement is \(\left(a^{\frac{1}{n}}\right)^{n}=a\). This is an equation involving exponentiation, and we need to check if this statement is true or false. In essence, we need to show that after taking the 'nth root' of a and then raising that to the 'n' power, we get back the original number a.

Step 2: Breaking down the given statement

The given statement is comprised of two operations: taking the nth root of a and then raising it to the n power. So to simplify the statement, first, we'll raise a to the power of \(\frac{1}{n}\), and then raise the result to the power of n. \[ \left(a^{\frac{1}{n}}\right)^{n} \]

Step 3: Using exponentiation rules

We can use the rule of exponentiation that says \((a^{m})^{n} = a^{m \times n}\). Using this rule for our problem, we get: \[\left(a^{\frac{1}{n}}\right)^{n} = a^{\frac{1}{n} \times n} \] Since \(\frac{1}{n} \times n = 1\), the above equation becomes: \[a^{1} = a\]

Step 4: Conclusion about the statement

We have shown that the given statement holds true for any natural number n and any positive real number a: \(\left(a^{\frac{1}{n}}\right)^{n} = a\). Therefore, the statement is true.

Key Concepts

Natural Numbers

Natural numbers are the numbers we often use when counting. These are the numbers starting from 1, 2, 3, and so on, continuing indefinitely. They do not include zero or any negatives. Here are a few simple points to remember about natural numbers:

• Natural numbers are always positive integers.
• Each natural number is followed by another natural number, forming an endless sequence.
• Natural numbers are used in basic counting "one, two, three..." and ordering "first, second, third..."

In the context of exponentiation, if a number is raised to the power of a natural number, it means we are multiplying the base by itself, the number of times indicated by the exponent. For example, if we have a number, say 2, and we want to raise it to the power of 3, written as 2^3, it simply means 2 multiplied by itself 3 times.

Positive Real Numbers

A positive real number is a broad term that covers any number greater than zero. This includes whole numbers, fractions, and decimals. Here are some key features of positive real numbers:

• They can be decimals, like 2.34 or fractions like 3/4.
• They are strictly greater than zero, located on the right side of zero on the number line.
• Use of positive real numbers is common in measurements, currency, statistics, and many other real-world scenarios.

When it comes to exponentiation with positive real numbers, we are dealing with the operation of raising these numbers to a certain power. Positive real numbers have the special capability to be raised even to fractional powers, which relates to taking roots, making them versatile in various mathematical operations and functions.

Exponentiation Rules

Exponentiation involves raising a number to a certain power. This operation is a key concept in mathematics, covering rules which make manipulation of exponents easier and clearer. Here are some fundamental exponentiation rules to keep in mind:

• For any positive number \(a \), raising it to the power \(0\) gives 1, i.e., \(a^0 = 1\).
• If you multiply the same base with different exponents, you can add the exponents: \(a^m \times a^n = a^{m+n}\).
• For division: if the same base has different exponents, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Power of a power rule: \((a^m)^n = a^{m \times n}\).

These rules are pivotal when simplifying expressions with exponents. For the specific concept of taking the nth root followed by raising it to the power of n, as discussed in the exercise, the rule \((a^{\frac{1}{n}})^n = a\) holds true due to the way exponents interact through multiplication. Understanding these basic rules helps in assessing and verifying statements in mathematics effectively.