Problem 7 - Elementary and Intermediate Algebra

Question

Fill in the blanks. a. The power rule for solving radical equations states that if \(x\), \(y,\) and \(n\) are real numbers and \(x=y,\) then\(x=y\) b. If \(\sqrt[n]{a}\) is a real number, then \((\sqrt[n]{a})^{n}=\)

Step-by-Step Solution

Answer

a. \(x^n = y^n\); b. \(a\).

Step 1: Understanding the Power Rule

We are given that for real numbers \(x\), \(y\), and \(n\), if \(x = y\), then applying the power rule, both sides can be raised to the power of \(n\) maintaining equality: \(x^n = y^n\). This is the completion for part (a).

Step 2: Understanding the Root and Exponent Relationship

For a real number \(a\), if we have the \(n\)th root of \(a\) as \(\sqrt[n]{a}\), then raising it to the power of \(n\) gives the original number: \((\sqrt[n]{a})^n = a\). This completes the answer for part (b).

Key Concepts

Power Rule

The power rule is an essential tool for solving radical equations. It allows us to manipulate equations by raising both sides to a power. This is valid under the condition that both sides of the equation start off equal. Here’s how it works:

Suppose you have an equation like \( x = y \). The power rule tells us that if you raise both sides to the power of \( n \) (where \( n \) is a real number), the equality remains, i.e., \( x^n = y^n \).

• This rule helps simplify and solve equations involving exponents and radicals.
• It's crucial in converting radical expressions into more manageable algebraic expressions.

Just remember that the power rule relies on both sides of the equation being initially equal.

Nth Root

The concept of the \( n \)th root is fundamental when dealing with powers and radicals. An \( n \)th root of a real number \( a \) is written as \( \sqrt[n]{a} \).

Here are the key points:

• The \( n \)th root essentially "undoes" what raising to a power does. For example, if \( b^n = a \), then \( \sqrt[n]{a} = b \).
• The operation \( (\sqrt[n]{a})^n = a \) demonstrates how the \( n \)th root and the power of \( n \) are inverse operations.

By understanding this root-exponent relationship, you can effectively tackle equations that contain both radicals and powers.

Real Numbers

Real numbers are a broad category of numbers that include all the numbers on the continuous number line. This category encompasses several types of numbers:

• Integers (both positive and negative, including zero)
• Fractions and decimals
• Irrational numbers like \( \pi \) and \( \sqrt{2} \), which cannot be expressed as exact fractions

Real numbers play a vital role in solving equations involving radicals and powers because the properties of real numbers ensure operations such as taking roots or raising to powers are valid. Understanding the nature of real numbers is foundational for applying mathematical rules to solve equations consistently and correctly.