Problem 17 - Algebra A Combined Function

Question

Use radical notation to rewrite each expression. Simplify if possible. $$ 16^{3 / 4} $$

Step-by-Step Solution

Answer

The expression simplifies to 8.

Step 1: Rewrite the Exponent Using Radical Notation

The expression given is \( 16^{3/4} \). The exponent \( \frac{3}{4} \) indicates that we can rewrite this expression as a radical. The denominator of the fraction \( 4 \) becomes the index of the radical, and the numerator \( 3 \) remains as the exponent: \( \sqrt[4]{16^3} \).

Step 2: Simplify Using Properties of Exponents

Instead of calculating \( 16^3 \) and then taking the fourth root, we can use the property of exponents to simplify: \( \sqrt[4]{16^3} = (\sqrt[4]{16})^3 \). So, we first need to simplify \( \sqrt[4]{16} \).

Step 3: Calculate the Fourth Root of 16

To simplify further, calculate the fourth root of 16. Since \( 16 = 2^4 \), \( \sqrt[4]{16} = 2 \).

Step 4: Final Simplification

Now substitute \( \sqrt[4]{16} = 2 \) back into the expression we have from Step 2: \( (\sqrt[4]{16})^3 = 2^3 \). Finally, calculate \( 2^3 = 8 \).

Key Concepts

Exponent Rules

Understanding exponent rules helps us handle expressions like \(16^{3/4}\). Exponents are a shorthand way of expressing repeated multiplication. For example, \(a^n\) means multiplying \(a\) by itself \(n\) times. When dealing with fractional exponents, the fraction's numerator and denominator play distinct roles.

• The numerator indicates the power to which the base is raised.
• The denominator tells the root to extract.

This means, in our expression \(16^{3/4}\), the base of 16 is raised to the 3rd power and then the fourth root is taken. Exploring such exponent rules gives us the option to rewrite expressions using radicals or keep them in exponential form.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and higher-order roots. In this scenario, we are working with fourth roots. To rewrite an expression like \(16^{3/4}\) using radical notation, we recognize that the exponent's denominator becomes the index of the radical.Thus, \(16^{3/4}\) can be rewritten as \(\sqrt[4]{16^3}\), where 4 is the index of the radical, indicating that we are taking the fourth root of \(16^3\). This helps us understand how to transition between exponential and radical forms, which is a key skill in algebra.

Simplifying Radicals

Simplifying radicals is all about making the expression easier to work with. Often the root of a number inside the radical can be simplified by factoring the number into its prime components.Consider the expression \(\sqrt[4]{16^3}\). We break it down by first calculating \(\sqrt[4]{16}\). Since \(16\) can be written as \(2^4\), it's easy to see that \(\sqrt[4]{16} = 2\). From here, we substitute back: \((\sqrt[4]{16})^3 = 2^3\), leading us to our simplified result: \(8\).This process not only simplifies the computation but also enhances our understanding of how radicals interact with exponents, making it simpler to handle in future problems.