Problem 64 - College Algebra

Question

Simplify each expression. $$\sqrt[5]{1,024 c^{10}}$$

Step-by-Step Solution

Answer

The simplified expression is \( 4c^2 \).

Step 1: Identify the Perfect Power Inside the Radical

The expression given is \( \sqrt[5]{1,024c^{10}} \). Notice that 1,024 is a perfect fifth power. We can express 1,024 as a power of 2: \( 1,024 = 2^{10} \). This is because \( 2^{10} = 1,024 \). Similarly, \( c^{10} \) is already in a perfect power form for fifth roots since \( c^{10} = (c^2)^5 \).

Step 2: Apply the Radical to Each Factor

We can split the expression inside the radical into its factors and take the fifth root of each: \( \sqrt[5]{1,024c^{10}} = \sqrt[5]{2^{10}} \cdot \sqrt[5]{c^{10}} \).

Step 3: Simplify Each Part Separately

Now, calculate each of the fifth roots: \[ \sqrt[5]{2^{10}} = 2^{10/5} = 2^2 = 4 \] \[ \sqrt[5]{c^{10}} = c^{10/5} = c^2 \]

Step 4: Combine the Results

Multiply the results of each separate fifth root: \( 4 \cdot c^2 \). So, \( \sqrt[5]{1,024c^{10}} = 4c^2 \).

Key Concepts

Understanding the Fifth Root

The fifth root of a number is an operation that is similar to squaring or taking the square root, but it is applied five times. When we talk about the fifth root of a number, we are looking for a value that, when raised to the fifth power, gives the original number. The notation for fifth root is often shown as \( \sqrt[5]{x} \).

For example, if we take the fifth root of 32, we refer to the number that satisfies \( x^5 = 32 \). In this case, the number is 2, because \( 2^5 = 32 \). Practically, the fifth root operation is comparable to division in the exponent world. It reduces the original exponent by a factor of five.

Identifying Perfect Powers

Perfect powers are numbers that can be expressed as an integer raised to another integer power. These are important in simplifying radical expressions, as recognizing them makes it easier to simplify.

For example, 1,024 is a perfect power because it can be written as \( 2^{10} \). Similarly, \( c^{10} \) is a perfect power because it can be expressed as \((c^2)^5 = c^{10} \).

Identifying these perfect powers successfully allows us to simplify radical expressions efficiently, as we can break them down into smaller, easier parts to work with.

Simplifying Radical Expressions

Radical expressions involve roots, like square roots or fifth roots. To simplify a radical expression, you often need to identify and extract any perfect power factors inside the radical.

Suppose we have the expression \( \sqrt[5]{1,024c^{10}} \). First, break it into parts as \( \sqrt[5]{2^{10}} \) and \( \sqrt[5]{c^{10}} \). By simplifying both parts, we find:

• \( \sqrt[5]{2^{10}} = 2^{10/5} = 2^2 = 4 \)
• \( \sqrt[5]{c^{10}} = c^{10/5} = c^2 \)

Finally, combine these results to obtain the simplified expression 4\( c^2 \). The process involves consistently applying root operations and reducing exponents through division.

Role of Exponents

Exponents are a shorthand way of expressing repeated multiplication. For instance, \( x^3 \) means \( x \times x \times x \). Understanding how to manipulate exponents is crucial in simplifying radical expressions, especially when taking roots.

In our example, \( \sqrt[5]{1,024c^{10}} \), we rewrite 1,024 and \( c^{10} \) in terms of exponents: \( 1,024 = 2^{10} \) and \( c^{10} = (c^2)^5 \). When finding the fifth roots, we divided the exponents by 5 thanks to the laws of exponents (\( a^{m/n} = \sqrt[n]{a^m} \)), simplifying to:

• \( \sqrt[5]{2^{10}} = 2^2 = 4 \)
• \( \sqrt[5]{c^{10}} = c^2 \)

So, knowing how to handle exponents efficiently aids in breaking down complex expressions into simpler forms.