Problem 109 - College Algebra Essentials

Question

In Exercises \(109-110\), evaluate each expression. $$\sqrt[3]{\sqrt[4]{16}+\sqrt{625}}$$

Step-by-Step Solution

Answer

The evaluated result of the expression \( \sqrt[3]{\sqrt[4]{16}+\sqrt{625}} \) is 3

Step 1: Evaluate \( \sqrt[4]{16} \)

The fourth root of 16 is 2 since \( 2^4 = 16 \). The value of \( \sqrt[4]{16} \) is therefore 2.

Step 2: Evaluate \( \sqrt{625} \)

The square root of 625 is 25 since \( 25^2 = 625 \). Therefore, the value of \( \sqrt{625}) \) is 25.

Step 3: Add the values

The next step is to add the values from Step 1 and Step 2. Therefore, \( 2 + 25 = 27 \).

Step 4: Evaluate \( \sqrt[3]{27} \)

The cube root of 27 is 3 since \( 3^3 = 27 \). The cube root of the sum from Step 3 hence results in 3.

Key Concepts

Roots of Numbers

Understanding the roots of numbers is like uncovering layers within a box. Imagine a number snug inside a box and each root is a key that unlocks one layer. When we talk about the square root, symbolized by \( \sqrt{...} \), we are looking for the number that, when multiplied by itself, gives us the original number inside the box - the square. On the other hand, cube roots, signified by \( \sqrt[3]{...} \), will tell us which number was used three times in multiplication to get the trapped number.

For example, if we're given a problem like \( \sqrt[4]{16} \) or \( \sqrt{625} \) from the exercise provided, we're peeling back four and two layers, respectively. The fourth root of 16 brings us down to 2 because \( 2 \times 2 \times 2 \times 2 = 16 \), and the square root of 625 takes us down to 25 since \( 25 \times 25 = 625 \). Visualizing these number layers can make it easier to understand why roots are essential building blocks in mathematics.

Exponentiation

Exponentiation is a way to express repeated multiplication. It's like chanting a number's name several times. In the mystical land of mathematics, if we have a number \( a \) and we want to multiply it by itself \( b \) times, we would write this as \( a^b \). Here, \( a \) is the base, the number being multiplied, and \( b \) is the exponent, which tells us how many times to echo the base.

In our example, \( 2^4 = 16 \) pronounces the number 2, four times in a row in a product (\( 2 \times 2 \times 2 \times 2 \)), and \( 3^3 = 27 \) echoes the number 3, thrice (\( 3 \times 3 \times 3 \)). This pattern of repeated multiplication helps us simplify and understand how bases and exponents work together to create powerful expressions.

Simplifying Expressions

Simplifying expressions is similar to cleaning up a room so that it's neat and tidy. You take a complex, cluttered expression and rearrange it, combine like terms, and reduce it to its cleanest form, giving you a clearer understanding of what's really there at a glance. It involves basic operations like addition, multiplication, and finding roots or exponents as needed.

Take our combined expression \( \sqrt[3]{\sqrt[4]{16}+\sqrt{625}} \). After calculating \( \sqrt[4]{16} = 2 \) and \( \sqrt{625} = 25 \), we add them together to get 27. What we have in the end is a neat and simplified expression \( \sqrt[3]{27} \), which is further simplified to 3. Going through these steps not only makes the solution clear to anyone who looks at it but also ensures a deeper understanding and an organized approach to tackling mathematical problems.