Problem 29 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ (\sqrt[3]{64})^{4} $$

Step-by-Step Solution

Answer

The result of \( (\sqrt[3]{64})^{4} \) is 256.

Step 1: Simplify the Cube Root

Begin by simplifying the cube root of 64. A cube root asks the question 'what number times itself twice gives me this?' In the case of 64, the cube root is 4, because \(4 * 4 * 4 = 64\). So, \( \sqrt[3]{64} = 4 \).

Step 2: Apply the Exponent

Now that we have simplified \( \sqrt[3]{64} \) to 4, we need to apply the exponent. The exercise asks us to raise this result to the 4th power, meaning we multiply it by itself four times. So, \(4^4 = 4 * 4 * 4 * 4 = 256\).

Key Concepts

Cube Root

A cube root helps us identify a number which, when multiplied by itself three times, gives us the original value. In simple terms, the cube root of a number like 64 asks, "What number times itself twice equals 64?" Here is how you can think about it:

• Imagine you have a number, say \( n \).
• When \( n \times n \times n = 64 \), \( n \) is the cube root of 64.
• For 64, the answer is 4 because \( 4 \times 4 \times 4 = 64 \).

Understanding cube roots helps simplify problems that involve finding a factor expressed as a base to the power of three. Recognizing this concept can aid in various thematically similar problems, providing a robust understanding of roots in general.

Simplifying Radicals

Simplifying radicals is about making expressions like roots easier to work with. When we encounter a radical expression, our goal is to express it in the simplest form possible. Take the cube root of 64, for example. The sequence goes:

• Find the number which when cubed gives 64, which we found is 4.
• Hence, \( \sqrt[3]{64} = 4 \).

Once the radical is simplified, it becomes much easier to apply further operations, such as exponentiation. This process of simplifying helps in evaluating expressions correctly and efficiently, providing a clear path to solutions without computation aid.

Exponent Rules

Exponent rules are foundational for solving problems involving powers. These rules dictate how we handle expressions that involve numbers raised to any power. Here's how it works in our context:

• After simplifying \( \sqrt[3]{64} \) to 4, you're asked to use an exponent.
• The expression \( 4^4 \) means multiplying 4 by itself four times: \( 4 \times 4 \times 4 \times 4 = 256 \).
• Exponents essentially provide a shorthand for repeated multiplication.

Grasping exponent rules helps in simplifying expressions and performing calculations more efficiently. Common rules include multiplying powers by adding their exponents or raising a power to another power by multiplying exponents. These rules are essential to work through any problem involving exponentiation.