Problem 107 - Algebra A Combined Function

Question

Find and correct the error. \(\frac{\sqrt[3]{64}}{\sqrt{64}}=\sqrt[3]{\frac{64}{64}}=\sqrt[3]{1}=1\)

Step-by-Step Solution

Answer

The value is \( \frac{1}{2} \), not 1.

Step 1: Evaluate the Numerator

First, evaluate the numerator of the fraction. The expression is \( \sqrt[3]{64} \). Since \( 64 = 4^3 \), we find that \( \sqrt[3]{64} = 4 \).

Step 2: Evaluate the Denominator

Next, evaluate the denominator of the fraction \( \sqrt{64} \). Since \( 64 = 8^2 \), we have \( \sqrt{64} = 8 \).

Step 3: Simplify the Fraction

Simplify the fraction \( \frac{\sqrt[3]{64}}{\sqrt{64}} \) using the values found in Step 1 and Step 2. Replace the terms with \( \frac{4}{8} \). Simplifying the fraction results in \( \frac{1}{2} \).

Step 4: Identify the Correction

The correction needed is to recognize the mistake at the step: \( \sqrt[3]{\frac{64}{64}} \). Since \( \frac{4}{8} = \frac{1}{2} \), the expression evaluates to \( \frac{1}{2} \), not 1. Replace \( \sqrt[3]{\frac{64}{64}} \) with \( \frac{1}{2} \).

Key Concepts

Cube Roots

Cube roots are an essential part of understanding more complex algebraic expressions. When we talk about the cube root of a number, we're looking for a special value that, when multiplied by itself three times, gives back the original number. For instance, the cube root of 64 is 4, because when you multiply 4 by itself three times (4 \( \times \) 4 \( \times \) 4), you get 64.

Some key points to remember about cube roots:

• A cube root is denoted by \( \sqrt[3]{} \).
• The cube root of a negative number is negative. For example, \( \sqrt[3]{-27} = -3 \), because \(-3 \times -3 \times -3 = -27 \).
• Unlike square roots, cube roots are defined for all real numbers, including negatives.

Understanding cube roots can help simplify more complex expressions, just like in the exercise you encountered. When working with cube roots, always break down the number into its smallest components or factors to determine the cube root easily.

Square Roots

The concept of square roots is fundamental for simplifying expressions and solving equations. A square root of a number is a value that, when multiplied by itself, gives the original number. Take the number 64, for example. The square root of 64 is 8, because 8 \( \times \) 8 equals 64.

You’ll often see square roots written as \( \sqrt{64} \). Here are some important facts about square roots:

• They are always non-negative in the context of real numbers. For instance, \( \sqrt{9} = 3 \), not -3.
• The square root of a perfect square is a whole number. That's why \( \sqrt{64} = 8 \).
• For non-perfect squares, square roots yield irrational numbers. For example, \( \sqrt{20} \approx 4.47 \).

Mastering square roots is essential for evaluating expressions accurately, as it helps simplify fractions with square roots in the numerator or denominator.

Evaluating Expressions

Evaluating mathematical expressions requires following a series of steps to find a numerical answer. The process involves operating on numbers, variables, and mathematical symbols to simplify or solve an equation.

When working with expressions like the original exercise, follow these general rules:

• Always break down the expression into simpler parts. Calculate each part before combining them.
• Start with evaluating roots or powers before proceeding to multiplication, division, and addition or subtraction.
• If it involves simplifying fractions, ensure all operations in both the numerator and the denominator are completed.

In the exercise example, evaluating \( \frac{\sqrt[3]{64}}{\sqrt{64}} \) involves calculating each root first and simplifying the fraction afterward. This detailed approach is crucial to avoid errors and reach correct conclusions, reinforcing your understanding of the concepts involved.