Math Problem Statement
simplfy ^4√32 over ^3√2
Solution
We need to simplify the expression:
[ \frac{\sqrt[4]{32}}{\sqrt[3]{2}} ]
Step 1: Express in Exponential Form
We rewrite the roots using exponents:
[ \sqrt[4]{32} = 32^{\frac{1}{4}}, \quad \text{and} \quad \sqrt[3]{2} = 2^{\frac{1}{3}} ]
Since , we substitute:
Thus, our expression becomes:
Step 2: Use Exponent Division Rule
Using the exponent subtraction rule:
we subtract the exponents:
Find a common denominator (LCM of 4 and 3 is 12):
Thus, we get:
Final Answer:
Would you like a deeper explanation or examples? Here are some related questions:
- How do you simplify ?
- How do exponent rules help in simplifying radical expressions?
- What are the properties of fractional exponents?
- How do you convert between radical and exponential form?
- How do you solve equations involving radicals and exponents?
Tip: Always express roots as fractional exponents to make simplifications easier!
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Math Problem Analysis
Mathematical Concepts
Exponents
Radicals
Fractional Exponents
Formulas
n√a = a^(1/n)
a^m / a^n = a^(m-n)
Theorems
Exponent Rules
Radical to Exponential Form Conversion
Suitable Grade Level
Grades 8-10