Problem 56 - Algebra 2

Question

Critical Thinking For what positive integers \(n\) is each of the statements true?. a. If \(x^{n}=b,\) then \(x\) is an \(n\) th root of \(b\) b. If \(x^{n}=b,\) then \(x=\sqrt[n]{b}\)

Step-by-Step Solution

Answer

Statement A is always true for all positive integers \(n\). For Statement B, the condition is true when \(n\) is a positive integer and the number \(x\) is also positive.

Step 1: Decipher Statement A

The condition in statement A: 'If \(x^{n}=b,\) then \(x\) is an \(n\)th root of \(b\)' is always true, irrespective of the value of 'n'. This follows from the standard definition of nth roots. If a number \(x^{n}\) equals \(b\), then \(x\) is referred to as an nth root of \(b\).

Step 2: Decipher Statement B

Statement B is quite similar, but has an extra condition: 'If \(x^{n}=b,\) then \(x=\sqrt[n]{b}\)'. While similar to the first statement, the condition in the second statement requires \(x\) to be specifically the principal or primary nth root of \(b\). By the definition of nth roots, there can be multiple nth roots for a given \(b\), especially for even values of \(n\). As a rule, for positive integers \(n\) greater than 2, \(x\) could be positive or negative, implying that there are two possible nth roots of \(b\). However, \(\sqrt[n]{b}\) refers only to the principal or primary nth root, which is always positive. So \(x=\sqrt[n]{b}\) is true only when \(n\) is a positive integer and \(x\) is positive. For negative \(x\), the statement would not hold true.

Key Concepts

Principal Nth Root

When we talk about the principal nth root, it means we are focusing on one specific root among all possible roots. For any positive integer , there can be multiple nth roots. However, we are usually interested in the principal one, which is always positive. This is particularly important when dealing with even integers. For instance, consider the equation \( x^2 = 4 \). Both \( x = 2 \) and \( x = -2 \) are solutions, but when we refer to \( \sqrt{4} \), it is precisely the positive root, \( 2 \).

• A principal nth root is always the non-negative solution when \( n \) is even.
• For odd \( n \), the principal root is simply the real root, which could either be positive or negative depending on \( b \).

Positive Integers

Positive integers are numbers like 1, 2, 3, and so forth, extending indefinitely. These are the numbers we often employ in both nth root calculations and nth power expressions, as they make the equations more straightforward. In the context of roots, they ensure the operations are real and manageable, since negative or non-integer roots could lead to complex or indeterminate results.

• In nth root scenarios, positive integers help specify the number of times a number is multiplied by itself to attain the given value.
• Only positive integers provide principal roots that are meaningful in typical real-number math.

Nth Power

The concept of nth power helps us understand how a number rises to a certain level within an expression. In mathematical terms, when a number \( x \) is raised to the power of \( n \), it is multiplied by itself \( n \) times. This is why exponents are so crucial; they compact the multiplication of numbers into a simple, easily-workable form.

• When \( x^n = b \), it implies \( x \times x \times ... \times x = b \) with \( x \) being repeated \( n \) times.
• An nth power calculation serves as the converse of finding a root.

Thus, understanding nth powers is directly linked to comprehending how nth roots operate, since the roots are fundamentally defined by reversing the power operation.

Root Calculation

Root calculation involves determining what number, when raised to the nth power, results in a given value. This is essentially the inverse operation of taking a power. For instance, calculating \( \sqrt[3]{8} \) means finding a number which when cubed (raised to the power of three) equals 8. The answer is 2, since \( 2^3 = 8 \).

• Calculating roots requires understanding both the concept of power and the definition of principal root especially for positive integers.
• Efficiency in root calculation often depends on familiarity with common powers and roots, such as squares (2nd roots) and cubes (3rd roots).

With strong clarity on root calculation, students can proficiently solve both simple and complex mathematical problems involving roots.