Problem 444 - Elementary Algebra

Question

In the following exercises, simplify. (a) \(\sqrt[3]{512}\) (b) \(\sqrt[4]{81}\) (c) \(\sqrt[5]{1}\)

Step-by-Step Solution

Answer

a) 8, b) 3, c) 1

Step 1: Simplify \(\sqrt[3]{512}\)

First, find the prime factorization of 512. We see that 512 can be expressed as powers of 2: \(\sqrt[3]{512} = \sqrt[3]{2^9}\). Since \(\sqrt[3]{2^9} = 2^{9/3} = 2^3 = 8\), we find that \(\sqrt[3]{512} = 8\).

Step 2: Simplify \(\sqrt[4]{81}\)

Next, find the prime factorization of 81: 81 can be written as \(\sqrt[4]{81} = \sqrt[4]{3^4}\). Since \(\sqrt[4]{3^4} = 3^{4/4} = 3\), we conclude that \(\sqrt[4]{81} = 3\).

Step 3: Simplify \(\sqrt[5]{1}\)

Finally, consider \(\sqrt[5]{1}\). Any non-zero number raised to any power is 1, so \(\sqrt[5]{1} = 1\).

Key Concepts

cube root

The cube root is a type of radical expression where we are looking for a number that, when multiplied by itself three times, results in the original number. For example, in the expression \(\sqrt[3]{512}\), we want a number that, when cubed, equals 512. To find this, we use prime factorization. Breaking down 512, we see it equals \(2^9\). Thus, \(\sqrt[3]{2^9} = 2^{9/3} = 2^3 = 8\). Therefore, \(\sqrt[3]{512} = 8\).

Here are some key points to remember:

• Cube root involves finding a number that, when cubed, gives the original number.
• Cube root of \(x\) is written as \(\sqrt[3]{x}\).
• Prime factorization can simplify the process.

Breaking the original number down into prime factors can make simplification easier.
Simply distribute the radical to each factor and simplify accordingly.

fourth root

The fourth root is another radical expression. This time, we look for a number which, when raised to the power of four, equals the original number. For example, \(\sqrt[4]{81}\) seeks a number that satisfies the equation \(x^4 = 81\).
Starting with the prime factorization of 81, we get \(3^4\). Therefore, \(\sqrt[4]{3^4} = 3^{4/4} = 3\). Thus, \(\sqrt[4]{81} = 3\).
Some critical observations:

• Fourth root involves finding a number that, when raised to the power of four, equals the original number.
• The fourth root of \(x\) is expressed as \(\sqrt[4]{x}\).
• Factorizing numbers into their prime components can greatly simplify the solution.

Always remember to break down the expression through factorization and then distribute the radical efficiently.

fifth root

The fifth root goes a step further. It is the number that, when raised to the power of five, results in the original number. For example, consider the expression \(\sqrt[5]{1}\). Here, we are finding the number that, when raised to the power of five, equals 1.
This is relatively straightforward since any non-zero number raised to any power of one remains one. So, \(\text{it simply means} \sqrt[5]{1} = 1\).
Key takeaways for fifth roots:

• Fifth root deals with finding a number that, when raised to the power of five, equals the original number.
• The fifth root of \(x\) is denoted as \(\sqrt[5]{x}\).
• Special cases like \(\sqrt[5]{1} = 1\) are often simpler and more direct since they involve basic number properties.

Recognizing these patterns can greatly assist in solving similar problems efficiently.