Problem 7 - 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

The power rule for radical equations implies if two expressions are equal, squaring both sides or applying any power to both sides maintains the equality. Hence, for real numbers \(x\) and \(y\), and integer \(n\), if \(x = y\), then \(x^n = y^n\). Therefore, the first blank is filled by "\(x^n = y^n\)."

Step 2: Evaluating an Nth Root Raised to the Nth Power

The expression \(\sqrt[n]{a}\) signifies the nth root of a number \(a\). When this is raised to the power \(n\), it simplifies to the original number since raising a root to its corresponding power effectively "cancels out" the root. Thus, \((\sqrt[n]{a})^n = a\). So, the second blank is filled by "\(a\)."

Key Concepts

Power Rule

The power rule is a foundational concept in algebra, particularly when working with equations involving radicals and exponents. It states that if two quantities are equal, then their corresponding powers will also be equal, assuming the same power is applied to both sides. For example, if you know that \(x = y\), applying the power rule would mean \(x^n = y^n\). This principle is crucial when you are called upon to eliminate radicals or solve nonlinear equations.

• The power rule allows you to manipulate equations by removing roots through exponentiation.
• It is particularly helpful for solving equations where the variable exists under a radicand.

Use this rule to simplify and solve radical equations efficiently by lifting the equality through powers. However, always be mindful of potential extraneous solutions that might emerge when both sides of an equation are raised to an exponent.

Nth Root

Understanding the concept of the nth root is vital when dealing with radical equations. The nth root of a number \(a\) is another number, which when raised to the power \(n\) gives \(a\). It is generally denoted as \(\sqrt[n]{a}\). For instance, the cube root of 8 is 2 because \(2^3 = 8\).

• An nth root can be thought of as the inverse operation of raising a number to a power.
• Mathematically, if \(b^n = a\), then \(b = \sqrt[n]{a}\).

When you raise an nth root to its own power, the result is simply the original value under the root, e.g., \((\sqrt[n]{a})^n = a\). This property is exceptionally useful for simplifying expressions in equations and is a powerful tool for resolving radical forms.

Exponents

Exponents represent repeated multiplication and are a fundamental component of algebra. They allow us to express large numbers succinctly and perform powerful mathematical manipulations. If \(b\) is a base number and \(n\) is the exponent, then \(b^n\) means \(b\) multiplied by itself \(n\) times.

• Exponents are helpful for simplifying expressions and solving equations involving large products.
• They are the flipside of roots, meaning they can help retrieve the base number from a root, as seen with nth roots.

The laws that govern exponents, such as the power of a power rule, assist in simplifying complex algebraic expressions and enable efficient computation. In radical equations, understanding and correctly applying the rules of exponents are essential for simplifying and finding solutions.