Math Problem Statement

Simplify the expression \( \frac{\sqrt[3]{24^4}}{24} \)

Solution

It looks like the expression in the image is:

244324\frac{\sqrt[3]{24^4}}{24}

Let me simplify this step by step:

Step 1: Simplifying the numerator

The numerator is 2443\sqrt[3]{24^4}, which is the cube root of 24424^4. This can be rewritten as:

2443=2443\sqrt[3]{24^4} = 24^{\frac{4}{3}}

Step 2: The full expression

Now, the expression becomes:

244324\frac{24^{\frac{4}{3}}}{24}

Since 24=24124 = 24^1, we can rewrite this as:

2443241=24431=2413\frac{24^{\frac{4}{3}}}{24^1} = 24^{\frac{4}{3} - 1} = 24^{\frac{1}{3}}

Step 3: Simplifying the result

The final result is:

2413=24324^{\frac{1}{3}} = \sqrt[3]{24}

So, the simplified form of the expression is:

243\boxed{\sqrt[3]{24}}

Would you like more details or have any questions?

Here are five related questions to explore further:

  1. What is the cube root of 24 approximated to two decimal places?
  2. How do you simplify expressions involving roots and exponents?
  3. What is the general rule for dividing powers with the same base?
  4. Can the expression 243\sqrt[3]{24} be simplified further into a decimal?
  5. How would you express 2453242\frac{\sqrt[3]{24^5}}{24^2}?

Tip: When dealing with roots and exponents, converting expressions into powers often makes simplification easier.

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Math Problem Analysis

Mathematical Concepts

Exponents
Roots
Simplification of Expressions

Formulas

Cube root of a power: \( \sqrt[3]{a^b} = a^{\frac{b}{3}} \)
Division of powers with the same base: \( \frac{a^m}{a^n} = a^{m-n} \)

Theorems

Laws of Exponents

Suitable Grade Level

Grades 9-12