Problem 20 - Intermediate Algebra

Question

Evaluate each of the following. For example, \(\sqrt{25}=5\). \(\sqrt[4]{16^{4}}\)

Step-by-Step Solution

Answer

The value is 16.

Step 1: Understand the Expression

The given expression is \( \sqrt[4]{16^{4}} \). This expression involves finding the fourth root of \( 16^{4} \).

Step 2: Simplify the Expression Inside

First, examine the inner expression, \( 16^{4} \). Our goal is to simplify it. The key is to recognize that \(16\) could be expressed as \(2^4\). Therefore, \(16^{4}\) simplifies to \((2^4)^4\).

Step 3: Calculate the Expression Inside

Calculate \((2^4)^4\). Using the power rule \((a^m)^n = a^{m \cdot n}\), this becomes \(2^{4 \times 4} = 2^{16}\).

Step 4: Apply the Fourth Root

Now, we need to find the fourth root of \(2^{16}\): \(\sqrt[4]{2^{16}}\). Using the property \(\sqrt[n]{a^m} = a^{m/n}\), we have \((2^{16})^{1/4} = 2^{16/4} = 2^4\).

Step 5: Simplify the Result

Finally, evaluate \(2^4\). Hence, \(2^4 = 16\).

Key Concepts

Exponents and Powers

Exponents and powers are fundamental concepts in algebra. An exponent is a small number written above and to the right of a base number, indicating how many times the base is multiplied by itself. For example, in the expression \(a^n\), \(a\) is the base, and \(n\) is the exponent. This means you multiply \(a\), \(n\) times.Understanding exponents is crucial, as they allow us to express large numbers more simply, and perform calculations more efficiently. A few key rules about exponents include:

• Product of Powers Rule: \(a^m \cdot a^n = a^{m+n}\), which states that when you multiply like bases, you add the exponents.
• Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\), meaning you multiply the exponents when raising a power to another power.
• Power of a Product Rule: \((ab)^n = a^n \cdot b^n\), which indicates that you apply the exponent to each factor in the product separately.

Remembering these rules can greatly simplify complex algebraic expressions, as they help break down calculations into more manageable steps.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form. This process often helps make calculations easier and clearer. When simplifying expressions with exponents, you apply rules such as the power of a power to combine terms and reduce the expression.A good strategy is to start by simplifying the expression inside any parentheses first, using known rules and properties. Next, apply any exponential rules such as converting bases or reducing coefficients. For example, it's often necessary to re express a number, like recognizing that 16 can be rewritten as \(2^4\), which allows for simplified calculations later on.Once you have simplified each part of the expression using the correct rules, the result should be a version of the expression that is easier to interpret and further compute if necessary. Simplifying well can significantly impact the efficiency and accuracy of solving problems.

Roots and Radicals

Roots and radicals are methods for finding the base of a power when you are given its exponentiated form. These are often represented in radical notation. The most common root is the square root, which is written as \(\sqrt{a}\), but roots can be of any order, like cube roots or fourth roots.Radicals express the inverse operation of raising a number to a power. For instance, finding the square root of \(x^2\) asks what number squared gives \(x^2\). The answer is simple: \(x\). Similarly, a fourth root, like \(\sqrt[4]{a}\), determines what number raised to the fourth power equals \(a\).In solving problems, you often use radical properties, such as:

• The radical of a product: \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\)
• Simplifying powers under a root: \(\sqrt[n]{a^m} = a^{m/n}\)

Understanding how to manipulate and simplify radicals is vital for efficiently solving higher order algebra problems.