Math Problem Statement

Express x√(x^7) in simplest radical form.

Solution

The expression provided in the image is xx7x \sqrt{x^7}, and the task is to simplify it to its simplest radical form.

Step-by-step solution:

  1. Simplifying the radical: x7=x6x=(x3)2x\sqrt{x^7} = \sqrt{x^6 \cdot x} = \sqrt{(x^3)^2 \cdot x} Since (x3)2=x3\sqrt{(x^3)^2} = x^3, we get: x7=x3x\sqrt{x^7} = x^3 \sqrt{x}

  2. Multiplying by the outside term: Now, we multiply this result by the original xx: xx3x=x4xx \cdot x^3 \sqrt{x} = x^4 \sqrt{x}

Thus, the expression simplifies to x4xx^4 \sqrt{x}.

Correct answer:

The correct answer is x4xx^4 \sqrt{x}, which matches option (C) in the image.

Would you like further details on this solution or have any questions?

Here are 5 related questions for deeper understanding:

  1. How do you simplify expressions with square roots involving exponents?
  2. Why can we factor powers of xx inside square roots?
  3. How would the simplification differ if it were a cube root instead of a square root?
  4. Can you simplify x10\sqrt{x^{10}} in a similar way?
  5. How do you handle mixed radical and exponential expressions in general?

Tip: Always look for perfect squares when simplifying square roots involving variables to simplify the expression more easily.

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Math Problem Analysis

Mathematical Concepts

Algebra
Radicals
Exponents

Formulas

√(a^b) = a^(b/2)

Theorems

Laws of Exponents

Suitable Grade Level

Grades 9-12