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 32=2532 = 2^5, we substitute:

3214=(25)14=25432^{\frac{1}{4}} = (2^5)^{\frac{1}{4}} = 2^{\frac{5}{4}}

Thus, our expression becomes:

254213\frac{2^{\frac{5}{4}}}{2^{\frac{1}{3}}}

Step 2: Use Exponent Division Rule

Using the exponent subtraction rule:

am÷an=amna^m \div a^n = a^{m-n}

we subtract the exponents:

5413\frac{5}{4} - \frac{1}{3}

Find a common denominator (LCM of 4 and 3 is 12):

54=1512,13=412\frac{5}{4} = \frac{15}{12}, \quad \frac{1}{3} = \frac{4}{12}

1512412=1112\frac{15}{12} - \frac{4}{12} = \frac{11}{12}

Thus, we get:

211122^{\frac{11}{12}}

Final Answer:

21112\mathbf{2^{\frac{11}{12}}}


Would you like a deeper explanation or examples? Here are some related questions:

  1. How do you simplify 64543\frac{\sqrt[5]{64}}{\sqrt[3]{4}}?
  2. How do exponent rules help in simplifying radical expressions?
  3. What are the properties of fractional exponents?
  4. How do you convert between radical and exponential form?
  5. 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