Problem 5 - College Algebra with Modeling and Visualization

Question

Simplify the expression. Assume that all variables are positive. $$ \sqrt[3]{4} \cdot \sqrt[3]{16} $$

Step-by-Step Solution

Answer

The expression simplifies to 4.

Step 1: Express the Radicals as Powers

Express each cube root as a power. We know that the cube root of a number \( a \) can be written as \( a^{1/3} \). Thus, \( \sqrt[3]{4} = 4^{1/3} \) and \( \sqrt[3]{16} = 16^{1/3} \).

Step 2: Use the Product Rule for Exponents

The expression \( \sqrt[3]{4} \cdot \sqrt[3]{16} \) can be rewritten using the identity \( a^{m} \cdot a^{n} = a^{m+n} \):\[ 4^{1/3} \cdot 16^{1/3} = (4 \cdot 16)^{1/3} \]

Step 3: Simplify Inside the Parentheses

Simplify the expression inside the parentheses:\[ 4 \cdot 16 = 64 \]

Step 4: Simplify the Cube Root

Now simplify the cube root of 64:\[ (64)^{1/3} \]Since \( 64 = 4^3 \), we can simplify the expression to:\[ (4^3)^{1/3} = 4^{3/3} = 4^1 = 4 \]

Step 5: Write the Final Simplified Result

The simplified form of the original expression is:\( 4 \).

Key Concepts

Cube Roots

Cube roots are one of the interesting ways in which we work with roots in mathematics. If you've ever wondered how to "unsquare" a cube, you're on the right track! A cube root is what you need to find a number which, when multiplied by itself three times (hence "cubed"), gives you the original number. For example, the cube root of 27 is 3 because when 3 is multiplied by itself twice more (3 x 3 x 3), we get 27.

We can represent the cube root of a number, say 8, as: \( \sqrt[3]{8} \). In terms of powers, that's equivalent to saying \( 8^{1/3} \). This is super handy because it makes cube roots compatible with the rules for exponents, allowing us to simplify expressions such as \( \sqrt[3]{4} \cdot \sqrt[3]{16} \).

In solving such problems, expressing these cube roots as powers helps avoid unnecessary complexity and makes any further calculations straightforward!

Product Rule for Exponents

The product rule for exponents is a powerful tool when simplifying expressions. It states that when you multiply two powers with the same base, you simply add the exponents. Mathematically, it's represented as follows: \( a^m \cdot a^n = a^{m+n} \).

This makes life much easier when dealing with expressions like \( 4^{1/3} \cdot 16^{1/3} \). Rather than getting bogged down in complexity, recognize that both parts have the same cube root power. Therefore, we can multiply the numbers inside the cube root first (like 4 and 16 in our example), and then apply the cube root principle to the result.

This elegant move simplifies the process significantly! By expressing the product inside the cube root as \( (4 \cdot 16)^{1/3} \), you are using the product rule for exponents cleverly. This is why mathematicians love clean and efficient rules like this!

Powers and Exponents

Powers and exponents form the backbone of a lot of mathematical operations. They enable us to shorthand repeated multiplication, such as \( 3^4 \) which means \( 3 \times 3 \times 3 \times 3 \). Understanding how to manipulate them is crucial for simplifying expressions.

An important concept is the reciprocal of powers, such as \( a^{1/3} \), which stands for the cube root of \( a \). This allows us to express roots in terms of exponents, providing versatility in mathematical operations. For instance, \( (64)^{1/3} \) can be simplified smoothly, especially when we recognize that \( 64 \) is actually \( 4^3 \).

This way, \( (4^3)^{1/3} \) simplifies all the way down to \( 4 \) because \( 4^{3/3} = 4^1 \) which is just \( 4 \). Realizing the equivalence between roots and fractional exponents opens many doors when simplifying and solving complex expressions!