Math Problem Statement
Solution
The problem is asking to explain why:
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).
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The cube root of 8 is:
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The cube root of 27 is:
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 simplifies to , which explains why:
Would you like more details on any specific step, or do you have any questions?
Here are 5 related questions to explore further:
- How would you simplify ?
- What is the cube root of a negative number, for example ?
- Can cube roots always be simplified to exact fractions like in this example?
- How does the process differ when finding a square root versus a cube root?
- What is the general rule for taking the -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