Problem 29 - Algebra 1

Question

Simplify the expression. $$\sqrt[3]{\frac{8}{64}}$$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt[3]{\frac{8}{64}}\) is \(\frac{1}{2}\)

Step 1: Simplify the Fraction

Simplify the fraction \(\frac{8}{64}\) to its simplest form. The simplified form of the fraction is \(\frac{1}{8}\). The equation now looks like \(\sqrt[3]{\frac{1}{8}}\).

Step 2: Solving the Cube Root

Now, the cube root of \(\frac{1}{8}\) needs to be solved. The cube root of 8 is 2, hence, the cube root of \(\frac{1}{8}\) is \(\frac{1}{2}\). The equation now simplifies to \(\frac{1}{5}\).

Key Concepts

Fraction Simplification

The process of fraction simplification involves converting a complex fraction into a more straightforward form without changing its value. It's an essential skill in algebra, making subsequent calculations easier and more accurate. To simplify a fraction, we look for a common factor in both the numerator (the top number) and the denominator (the bottom number). In our example, \( \frac{8}{64} \) can be simplified by recognizing that both 8 and 64 are divisible by 8. After dividing both by this common factor, we get \( \frac{1}{8} \) which is the fraction in its simplest form. This step is crucial because it lays the foundation for further operations such as cube root calculations, and it ensures clarity in evaluating radical expressions.

Cube Root Calculations

Cube root calculations are a bit like detective work where we're looking for the number that, when multiplied by itself three times, gives us the original value. In mathematical terms, the cube root of a number x is a number y such that \( y^3 = x \). When we're given the task to find the cube root of a fraction, we approach each part separately. If we look at our simplified fraction, \( \frac{1}{8} \) from the earlier step, the cube root of 1 is 1, since \( 1^3 = 1 \), and the cube root of 8 is 2, because \( 2^3 = 8 \). Therefore, the cube root of \( \frac{1}{8} \) is \( \frac{1}{2} \), simplifying our expression to its final form.

Radical Expressions

Understanding Radicals

The concept of 'radical expressions' comes from the idea of extracting roots, such as square roots, cube roots, and so on. These expressions contain a radical symbol — \( \sqrt{} \) for square roots and \( \sqrt[3]{} \) for cube roots, for example. Simplifying a radical expression means breaking it down into simpler parts that are easier to work with. This involves identifying and extracting perfect squares, cubes, or higher powers when dealing with more complex roots. In the given exercise, simplifying the cube root of a fraction is a matter of applying this concept to both the numerator and the denominator, making the expression more straightforward and manageable.

Algebraic Expressions

Algebraic expressions are phrases in the language of mathematics that can contain numbers, variables, and operation symbols. These expressions don’t have an equality sign like equations do, but they can still convey meaningful relationships and values. Simplifying algebraic expressions is a crucial algebra skill. It involves condensing the expression into the most efficient form possible, often by combining like terms, simplifying fractions, or calculating radicals. In this context, our initial expression \( \sqrt[3]{\frac{8}{64}} \) is an algebraic expression that involves both fraction simplification and cube root calculation. The goal is to rewrite it as simply as possible, which, through the steps taken, ends up being \( \frac{1}{2} \), a much more digestible form.