Problem 64 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\sqrt[5]{1,024 c^{10}}\)

Step-by-Step Solution

Answer

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

Step 1: Simplify the Numerical Part

Start by simplifying the numerical part under the fifth root. The number 1,024 can be broken down into prime factors as follows: \[1,024 = 2^{10}\]Since the expression involves the fifth root, see how the powers correspond to the root.

Step 2: Apply the Fifth Root to the Numerical Part

Take the fifth root of \(2^{10}\):\[\sqrt[5]{2^{10}} = (2^{10})^{1/5} = 2^{10/5} = 2^{2} = 4\]Thus, the result of the numerical part is 4.

Step 3: Simplify the Variable Part

Now consider the variable part \(c^{10}\):Apply the fifth root to the variable expression:\[\sqrt[5]{c^{10}} = (c^{10})^{1/5} = c^{10/5} = c^2\]Thus, the simplified expression of the variable part is \(c^2\).

Step 4: Combine the Simplified Parts

Combine both simplified parts from Steps 2 and 3:The expression \(\sqrt[5]{1,024c^{10}}\) simplifies to:\[4c^2\]

Key Concepts

Fifth Root

When we talk about the fifth root, we mean finding a number which, when multiplied by itself five times, gives the original number. In simpler terms, if you have a number \(a\) and you're finding the fifth root of some value \(b\), then \(a^5 = b\). The fifth root is written as \(\sqrt[5]{b}\).
For example, consider the number 32. The fifth root of 32 is 2, because \(2 \times 2 \times 2 \times 2 \times 2 = 32\). So, \(\sqrt[5]{32} = 2\).
To further illustrate, in our problem, we took the fifth root of \(2^{10}\). We transformed it as follows: \(\sqrt[5]{2^{10}} = (2^{10})^{1/5}\). This calculation involves transforming the expression into a power raised to the fifth root, which allowed us to simplify it to \(2^{2} = 4\).
Understanding roots and breaking down powers will make complex calculations much easier, especially when dealing with bigger numbers or expressions.

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks—its prime numbers. A prime number is a number greater than 1 that is only divisible by 1 and itself. When you write a number as a product of primes, you've completed its prime factorization.
For example, the number 1,024 was broken down into \(2^{10}\). This means we multiplied the number 2 by itself ten times to get 1,024. Here’s how it's done:

• 1,024 is even, so divide by 2. Keep dividing by 2 each time until the quotient is 1.
• You'll do this a total of ten times, which shows \(2^{10} = 1,024\).

Prime factorization makes it much easier to handle roots and powers because it breaks complex numbers into simple components. When simplifying roots, like with the fifth root, knowing the prime factorization helps you work out the exact value easily.

Exponent Rules

Exponent rules or laws are crucial in transforming and simplifying mathematical expressions. When you have expressions involving powers, such as \(c^{10}\) in our example, these rules come in handy.
Some basic exponent rules are:

• Product of Powers Rule: When multiplying like bases, add their exponents: \(a^m \cdot a^n = a^{m+n}\).
• Power of a Power Rule: To raise a power to another power, multiply the exponents: \( (a^m)^n = a^{m \cdot n} \).
• Quotient of Powers Rule: When dividing like bases, subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).

In solving \(\sqrt[5]{c^{10}}\), we utilized the power of a power rule by calculating \((c^{10})^{1/5} = c^{10/5} = c^2\). This simplified our expression dramatically. Mastery of exponent rules allows you to efficiently tackle complex expressions by manipulating powers with ease.