Problem 64 - College Algebra

Question

For the following exercises, simplify each expression. $$ \sqrt[3]{1,024 c^{11}} $$

Step-by-Step Solution

Answer

The simplified form is \( 8c^3 \times \sqrt[3]{2c^2} \).

Step 1: Identify the Cube Root

We need to simplify the expression \( \sqrt[3]{1,024 c^{11}} \). This is a cube root problem, so we'll simplify under the cube root by breaking down the number 1,024 and the variable \( c^{11} \).

Step 2: Prime Factorization of 1,024

Let's determine the prime factorization of 1,024. Divide by 2 repeatedly: \[ 1,024 \div 2 = 512, \quad 512 \div 2 = 256, \quad 256 \div 2 = 128, \quad 128 \div 2 = 64, \quad 64 \div 2 = 32, \quad 32 \div 2 = 16, \quad 16 \div 2 = 8, \quad 8 \div 2 = 4, \quad 4 \div 2 = 2 \]So, \( 1,024 = 2^{10} \).

Step 3: Rewrite the Expression

Now, replace 1,024 with its prime factorization in the cube root and write \( c^{11} \) as \( c^{9} imes c^{2} \): \[ \sqrt[3]{1,024 \, c^{11}} = \sqrt[3]{2^{10} \, c^{9} \, c^{2}} \]

Step 4: Apply the Cube Root Simplification

The cube root of a number is found by dividing the exponents by 3. Therefore:- \( \sqrt[3]{2^{10}} = 2^{10/3} = 2^3 \times 2^{1/3} \)- \( \sqrt[3]{c^{9}} = c^{9/3} = c^3 \)- \( \sqrt[3]{c^2} = c^{2/3} \) remainsThus, the simplified expression inside cube root becomes:\[ 2^3 \, c^3 \, \times \sqrt[3]{2^{1} \, c^2} \]

Step 5: Final Simplification

Simplify the expression by combining factors:\[ 2^3 \, c^3 = 8 \, c^3 \]So the entire simplified expression is:\[ 8c^3 \times \sqrt[3]{2 \, c^2} \]

Key Concepts

Prime Factorization

Prime factorization is the method of representing a number as a product of its prime factors. To simplify expressions with radicals, such as cube roots, knowing the prime factorization of a number can be very helpful. In our example, we started with the number 1,024. By repeatedly dividing by 2, a prime number, we found:

• 1,024 divided by 2 repeatedly until we reach 2.
• This resulted in 1,024 being equal to \( 2^{10} \).

Understanding prime factorization allows us to break down numbers into their simplest building blocks and aids in simplifying expressions under a radical, making it easier to apply exponent rules. This breakdown helps make algebraic computations straightforward, especially when simplifying cube roots or higher radicals. By expressing 1,024 as \( 2^{10} \), it becomes manageable to handle within the cube root operation.

Exponent Rules

Exponent rules are essential tools in algebra that help simplify expressions. They define how to handle mathematical operations involving powers or exponents. These rules apply not only to numbers but also to variables and expressions they power. In the context of our original exercise:

• We converted 1,024 into \( 2^{10} \) based on its prime factors.
• We transformed \( c^{11} \) into \( c^9 \times c^2 \).

When dealing with a cube root, the rules of exponents help us divide these powers by three. For instance, \( \sqrt[3]{2^{10}} \) becomes \( 2^{10/3} \). The same division applies to the exponents of variables. Understanding these rules aids in breaking down complicated expressions into more comprehensible forms, leading to further simplification.

Algebraic Expression Simplification

Algebraic expression simplification involves rewriting expressions in a simpler or more reduced form without changing their value. This process often uses a combination of mathematical operations such as combining like terms, employing exponent rules, and factoring. In our case:

• We expressed the cube root \( \sqrt[3]{1,024 \, c^{11}} \) as \( \sqrt[3]{2^{10} \, c^{9} \, c^{2}} \).
• This enabled us to break down the expression using prime factorization and exponent division.

Simplifying algebraic expressions is critical for solving equations more efficiently and understanding deeper math topics. It is often a prerequisite for further operations like finding values, solving equations, or integrating more advanced algebraic concepts.

Polynomial Expressions

Polynomial expressions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. Understanding and working with polynomials is vital for solving many types of algebraic problems.

• In the exercise, expressions like \( c^{9} \) and \( c^2 \) can appear as terms within polynomial expressions.
• Simplifying these helps in streamlining complex polynomial operations.

This type of simplification prepares the groundwork not only for basic polynomial operations but also for more advanced topics like factorization, solving polynomial equations, and understanding polynomial functions. Mastery of polynomial expressions is a gateway to deeper mathematical understanding and higher-level problem solving.