Math Problem Statement

Solution

The expression shown in the image is:

484\sqrt[4]{48}

This represents the 4th root (or the fourth root) of 48. To simplify or evaluate this expression, we can look for factors of 48 that are perfect fourth powers. However, since 48 does not have any factors that are perfect fourth powers (such as 1,16,81,1, 16, 81, \ldots), we will instead simplify it by factoring out any perfect squares (because the fourth root can be broken into square roots).

Simplifying 484\sqrt[4]{48}

First, let's factor 48 into its prime factors: 48=24×348 = 2^4 \times 3

Now, apply the 4th root to this factorization: [ \sqrt[4]{48} = \sqrt[4]{2^4 \times 3} ]

We can separate the expression into two parts, the 4th root of 242^4 and the 4th root of 3: [ \sqrt[4]{48} = \sqrt[4]{2^4} \times \sqrt[4]{3} ]

Since 244=2\sqrt[4]{2^4} = 2 (because taking the 4th root of a 4th power returns the base), we have: [ \sqrt[4]{48} = 2 \times \sqrt[4]{3} ]

Therefore, the simplified form of 484\sqrt[4]{48} is:

2342 \sqrt[4]{3}

Would you like further details or have any questions about this simplification?

Here are 5 related questions to consider:

  1. What are the rules for simplifying radicals and roots?
  2. How do we simplify higher-order roots like cube roots or fourth roots?
  3. Can we always find a simplified form for roots of non-perfect powers?
  4. How do we convert a radical expression to exponential form?
  5. What is the difference between simplifying a radical and evaluating a radical?

Tip: Breaking down a number into its prime factors is often a good strategy to simplify radical expressions.

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Math Problem Analysis

Mathematical Concepts

Roots and Radicals
Prime Factorization

Formulas

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Theorems

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Suitable Grade Level

Grades 9-12