Problem 107 - Algebra A Combined Function

Question

Find and correct the error. See the Concept Check in this section. $$ \frac{\sqrt[3]{64}}{\sqrt{64}}=\sqrt[3]{\frac{64}{64}}=\sqrt[3]{1}=1 $$

Step-by-Step Solution

Answer

The corrected value is \(\frac{1}{2}\).

Step 1: Identify the Expression

The given expression is \(\frac{\sqrt[3]{64}}{\sqrt{64}}\). This expression needs to be evaluated correctly as per mathematical rules.

Step 2: Simplify the Numerator and Denominator

Calculate each part of the fraction separately: \(\sqrt[3]{64} = 4\) because \(4^3 = 64\), and \(\sqrt{64} = 8\) because \(8^2 = 64\).

Step 3: Formulate the Correct Fraction

With the simplified values, the expression \(\frac{\sqrt[3]{64}}{\sqrt{64}}\) becomes \(\frac{4}{8}\).

Step 4: Simplify the Fraction

Simplify \(\frac{4}{8}\) by dividing the numerator and the denominator by their greatest common divisor, which is 4: \(\frac{4 \div 4}{8 \div 4} = \frac{1}{2}\).

Key Concepts

Cube Roots

Cube roots allow us to determine the number that, when multiplied by itself three times, gives us the original number. For example, the cube root of 64 is 4, because when you multiply 4 by itself three times (4 x 4 x 4), it equals 64. Cube roots are denoted as \( \sqrt[3]{\text{number}} \). This notation helps identify that it's a cube root calculation. Calculating cube roots can be achieved through various methods such as prime factorization and using a calculator when the numbers become large or complicated.To better understand cube roots, here are a few points:

• Simpler cube roots, such as \( \sqrt[3]{1} \), will always result in 1 because 1 multiplied by itself is always 1, regardless of the exponent.
• Cube roots grow larger as the original number increases, for instance, \( \sqrt[3]{8} = 2 \).

Square Roots

Square roots find the number that, when multiplied by itself, gives the original number. Square roots are denoted by the radical symbol \( \sqrt{} \). The square root of 64 is 8, because 8 multiplied by 8 returns to 64 again.For better grasping square roots, consider these useful tips:

• Each positive number actually has two square roots: a positive and a negative. For 64, they are +8 and -8, because both satisfy 8 x 8 and -8 x -8 equal 64.
• When simplifying expressions, it's common to focus on the principal square root, which is the positive square root value.

Using square roots consistently in solving equations or problems adds a layer of understanding mathematics in real-world applications such as area calculation of squares.

Simplifying Fractions

Simplifying fractions is a fundamental skill in algebra and mathematics in general. It involves reducing a fraction to its simplest form, where the numerator and the denominator have no common factors other than 1.Let's look at the simplified form of the fraction \( \frac{4}{8} \). Here, both 4 and 8 can be divided by their greatest common divisor, which is 4. So when we divide both the numerator and the denominator by 4, we find \( \frac{4}{8} = \frac{1}{2} \).Useful strategies to simplify fractions include:

• Identify the greatest common divisor of the numerator and the denominator.
• Divide both top and bottom by this divisor to reduce the fraction.
• Sometimes, factorizing both numerator and denominator can help you quickly see shared factors.

This process helps prevent errors in algebraic manipulations and ensures results express the simplest form of any fractional values.

Error Correction in Mathematics

Spotting and correcting mistakes is a crucial part of learning mathematics, ensuring you understand not just how to find an answer but why that answer is correct.Taking the original calculation \( \frac{\sqrt[3]{64}}{\sqrt{64}} \), a common error was simplifying directly to \( \sqrt[3]{\frac{64}{64}} \), which incorrectly leads to 1. Understanding how and why errors happen helps you avoid pitfalls:

• Always simplify expressions individually — for cube and square roots separately in any given expression.
• Re-evaluate steps if a solution seems incorrect or illogical. Mathematics is often verifying work for accuracy.
• Seek peer or instructor insights; discussing mistakes often solidifies understanding.

By developing a careful and methodical approach, correcting errors becomes an opportunity to deepen your comprehension and accuracy in mathematics.