Math Problem Statement

Explain why the cube root of 8/27 is 2/3.

Solution

The problem is asking to explain why:

8273=23\sqrt[3]{\frac{8}{27}} = \frac{2}{3}

Let's break it down step by step:

Step 1: Simplifying the Cube Root

We are tasked with taking the cube root of a fraction. The cube root of a fraction is the same as taking the cube root of the numerator and the denominator separately:

[ \sqrt[3]{\frac{8}{27}} = \frac{\sqrt[3]{8}}{\sqrt[3]{27}} ]

Step 2: Cube Roots of the Numerator and Denominator

Now, let's calculate the cube roots of the numerator (8) and the denominator (27).

  • The cube root of 8 is: 83=2(since 23=8)\sqrt[3]{8} = 2 \quad \text{(since } 2^3 = 8\text{)}

  • The cube root of 27 is: 273=3(since 33=27)\sqrt[3]{27} = 3 \quad \text{(since } 3^3 = 27\text{)}

Step 3: Form the Fraction

Now, substitute the cube roots back into the fraction:

[ \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3} ]

Thus, the cube root of 827\frac{8}{27} simplifies to 23\frac{2}{3}, which explains why:

8273=23\sqrt[3]{\frac{8}{27}} = \frac{2}{3}

Would you like more details on any specific step, or do you have any questions?

Here are 5 related questions to explore further:

  1. How would you simplify 641253\sqrt[3]{\frac{64}{125}}?
  2. What is the cube root of a negative number, for example 83\sqrt[3]{-8}?
  3. Can cube roots always be simplified to exact fractions like in this example?
  4. How does the process differ when finding a square root versus a cube root?
  5. What is the general rule for taking the nn-th root of a fraction?

Tip: Remember that when simplifying cube roots, you can always apply the cube root to the numerator and denominator separately.

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Math Problem Analysis

Mathematical Concepts

Cube Roots
Fractions
Simplification

Formulas

Cube root of a fraction: \( \sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}} \)
Cube root properties: \( \sqrt[3]{x^3} = x \)

Theorems

Cube Root Simplification

Suitable Grade Level

Grades 7-9