Problem 16 - Algebra A Combined Function

Question

Use radical notation to rewrite each expression. Simplify if possible. $$ (-32)^{1 / 5} $$

Step-by-Step Solution

Answer

\((-32)^{1/5}\) simplifies to \(-2\).

Step 1: Understand Radical Notation

Radical notation involves writing an expression using a radical symbol. The general form is \(a^{1/n}=\sqrt[n]{a}\), where \(n\) is the root. In this exercise, you need to rewrite \((-32)^{1/5}\) using radical notation.

Step 2: Apply Radical Notation

Convert \((-32)^{1/5}\) into radical form by using the rule \(a^{1/n}=\sqrt[n]{a}\). Thus, \((-32)^{1/5}\) becomes \(\sqrt[5]{-32}\).

Step 3: Simplify the Radical Expression

To simplify \(\sqrt[5]{-32}\), consider whether there exists a whole number \(x\) such that \(x^5 = -32\). Since \((-2)^5 = -32\), we can simplify \(\sqrt[5]{-32}\) to \(-2\).

Key Concepts

Fifth Root

The fifth root of a number is similar to the square root, but instead of multiplying a number by itself two times, you multiply it five times to achieve the original value. When talking about radical notation involving fifth roots, we use the expression \( \sqrt[5]{a} \). This notation tells us that we're looking for a number, which when raised to the power of five, gives us \( a \).

• Fifth root is the inverse operation of raising a number to the fifth power.
• To express a fifth root using exponents, we write \( a^{1/5} \).
• Finding the fifth root of a number can help simplify complex expressions.

In the example \((-32)^{1/5}\), we're tasked with finding a number, that when multiplied by itself five times equals \(-32\). For instance, \(-2\) is that number since \((-2)^5 = -32\).
Thus, \((-32)^{1/5}\) can be simplified to be \(-2\).

Simplification of Radicals

Simplifying radicals involves reducing a radical expression to its simplest form. When you simplify a radical like \( \sqrt[5]{-32} \), you're finding the smallest whole number that can be expressed as that root.

• Simplification helps make expressions easier to work with, especially in equations.
• You simplify a radical by finding common factors that can "come out" of the radical.
• In our case, the process involves recognizing patterns from powers of integers.

For \( \sqrt[5]{-32} \), consider whether \( -32 \) can be expressed as a power of a smaller integer. Since \((-2)^5 = -32\), we can simplify \( \sqrt[5]{-32} \) to \(-2\). This simplification step allows for easier computation and understanding of equations involving radicals.

This step is crucial, primarily when dealing with higher powers, as it results in a clearer and more manageable number.

Exponents and Radicals

Exponents and radicals are closely linked as they represent inverse operations. While exponents denote repeated multiplication, radicals signify finding the root of that multiplication.

• An expression like \( a^{1/n} \) can be rewritten as \( \sqrt[n]{a} \).
• Understanding both concepts is key to manipulating complex expressions in algebra.
• Both radicals and exponents can simplify expressions and solve equations.

In the exercise \((-32)^{1/5}\), the exponent \(1/5\) signals the fifth root, transforming it into radical notation as \( \sqrt[5]{-32} \).
Recognizing the relationship between these two allows you to freely convert between forms and simplify expressions accurately. For example, knowing the exponent form helped us rewrite the radical, leading to clarity in determining \( (-2) \) as the simplest form of the expression. This connection highlights the versatility and power of understanding both exponents and radicals.