Problem 496 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\sqrt[3]{128}+\sqrt[3]{250}\) (b) \(\sqrt[5]{729}+\sqrt[5]{96}\)

Step-by-Step Solution

Answer

The simplified form is: (a) \(2^{7/3} + 2^{1/3} \cdot 5\) (b) \(3^{6/5} + 2 \cdot 3^{1/5}\)

Step 1: Write the expression

The given exercises are: (a) \(\sqrt[3]{128} + \sqrt[3]{250}\) (b) \(\sqrt[5]{729} + \sqrt[5]{96}\)

Step 2: Simplify individual cube roots

First, find the cube roots of 128 and 250: \[128 = 2^7 \Rightarrow \sqrt[3]{128} = \sqrt[3]{2^7} \Rightarrow \sqrt[3]{2^7} = (2^7)^{1/3} = 2^{7/3}\] \[250 = 2 \times 5^3 \Rightarrow \sqrt[3]{250} = \sqrt[3]{2 \cdot 5^3} \Rightarrow \sqrt[3]{2 \cdot 5^3} = 2^{1/3} \cdot 5\]

Step 3: Write simplified form of cube roots

Using the results from Step 2, the simplified form of the original expression is: \[\sqrt[3]{128} + \sqrt[3]{250} = 2^{7/3} + 2^{1/3} \cdot 5 \]

Step 4: Simplify individual fifth roots

Next, find the fifth roots of 729 and 96: \[729 = 3^6 \Rightarrow \sqrt[5]{729} = \sqrt[5]{3^6} \Rightarrow \sqrt[5]{3^6} = (3^6)^{1/5} = 3^{6/5}\] \[96 = 2^5 \cdot 3 \Rightarrow \sqrt[5]{96} = \sqrt[5]{2^5 \cdot 3} \Rightarrow \sqrt[5]{2^5 \cdot 3} = 2 \cdot 3^{1/5} \]

Step 5: Write simplified form of fifth roots

Using the results from Step 4, the simplified form of the original expression is: \[\sqrt[5]{729} + \sqrt[5]{96} = 3^{6/5} + 2 \cdot 3^{1/5} \]

Key Concepts

cube roots

Cube roots are used to determine what number, when multiplied by itself three times, gives the original number. For example, when you take the cube root of 8, you are looking for a number which, when cubed, equals 8. The cube root of 8 is 2, since \(2 \cdot 2 \cdot 2 = 8\). Understanding cube roots is crucial in simplifying expressions involving radicals.

fifth roots

Fifth roots work similarly to cube roots, but here, you need to find a number that, when multiplied by itself five times, results in the original number. For example, the fifth root of 32 can be determined as follows: \(2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32\). This means the fifth root of 32 is 2. Simplifying expressions with higher-order roots, such as the fifth root, often involves breaking down numbers into their prime factors, making it easier to work with them in expressions.

expressions

Expressions involving radicals can seem challenging at first, but with practice, they become more intuitive. In exercises like these, it’s essential to break down each component step-by-step. Let's take the expression \(\sqrt[3]{128} + \sqrt[3]{250}\). By breaking down the terms, we can use properties of exponents to simplify each part. When you encounter terms like \(128 = 2^7\), you can use fractional exponents, \( (2^7)^{1/3} = 2^{7/3} \), to simplify the expression. Similarly, understanding how to deal with fifth roots and converting them using prime factorization helps simplify and solve the overall expression. Always ensure to address each part methodically for efficient problem-solving.