Problem 4 - Algebra 1: Concepts and Skills

Question

Evaluate the expression without using a calculator. $$ (\sqrt[3]{8})^{5} $$

Step-by-Step Solution

Answer

The evaluated expression \((\sqrt[3]{8})^{5}\) equals 32.

Step 1: Calculate the Cube Root

Firstly, we need to calculate the cube root of 8. The cube root of a number \(x\) is a value \(a\) which when cubed (multiplied to itself 3 times) gives the original number \(x\). We know that \(2 * 2 * 2 = 8\), thus \(\sqrt[3]{8} = 2\).

Step 2: Raise the Result to the Power of 5

Now, we need to raise the result from the previous step, which is 2, to the power of 5 (i.e., calculated as \(2^{5}\)). When we say \(2^{5}\), it means that we'll multiply 2 by itself five times. Hence, \(2^{5} = 2 * 2 * 2 * 2 * 2 = 32\).

Key Concepts

Cube Root

A cube root is a special type of root used in mathematics to find which number, when multiplied by itself two more times (making it a total of three times), gives the original number. For example, if we take the number 8, its cube root is 2 because multiplying 2 three times (2 * 2 * 2) equals 8.

To express this mathematically, we use the symbol \( \sqrt[3]{} \). Thus, \( \sqrt[3]{8} = 2 \)."

Understanding cube roots is essential as they frequently appear in algebra and geometry. Often, you'll need to simplify expressions involving cube roots to solve more complex problems.

Powers

Powers are a way to express how many times a number, called the base, is multiplied by itself. When you have \( 2^{5} \), the base is 2 and the power, also known as the exponent, is 5.

This mathematical notation is both a time-saver and a space-saver. Instead of writing \( 2 \times 2 \times 2 \times 2 \times 2 \), you write \( 2^{5} \). It shows that 2 is multiplied by itself five times. The outcome of \( 2^{5} \) is 32, meaning that 2, used as a factor five times, equals 32.

Powers are especially useful when dealing with very large or very small numbers, simplifying the notation and easing calculations.

Simplifying Expressions

Simplifying expressions means making a complex mathematical expression more understandable or easier to work with by reducing it to its simplest form.

By calculating cube roots, like \( \sqrt[3]{8} = 2 \), and then raising them to a power \( (2^{5} = 32) \), you're effectively simplifying a complex expression into a simple numerical result.

Simplified expressions make solving math problems more efficient and are often easier to understand and communicate. This makes them a crucial part of mathematical problem-solving and analysis. Decision-making in mathematics often relies on fully simplified expressions to ensure accuracy and clarity.