Math Problem Statement

Simplify the expression \(\sqrt{81 y^{21}}\), assuming y represents a positive real number.

Solution

The expression provided is:

81y21\sqrt{81 y^{21}}

We are asked to simplify this expression, assuming yy is a positive real number.

Step-by-Step Simplification:

  1. Rewrite the square root: 81y21=81y21\sqrt{81 y^{21}} = \sqrt{81} \cdot \sqrt{y^{21}}

  2. Simplify the square root of 81: 81=9\sqrt{81} = 9

  3. Simplify the square root of y21y^{21}: Using the property an=an/2\sqrt{a^n} = a^{n/2}, we can simplify: y21=y21/2=y10.5\sqrt{y^{21}} = y^{21/2} = y^{10.5}

  4. Final Simplified Expression: 9y10.59y^{10.5}

Thus, the simplified form is:

9y10.59y^{10.5}

Would you like further details on any step, or do you have any additional questions?

Related Questions:

  1. How do you simplify expressions involving higher roots?
  2. What happens if yy is not a positive real number in this case?
  3. Can you express y10.5y^{10.5} using radicals instead of exponents?
  4. How do you handle fractional exponents in algebraic expressions?
  5. What if the base 8181 were not a perfect square, how would you simplify the expression?

Tip:

When simplifying square roots, it's useful to remember that fractional exponents can often make the process more straightforward.

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Math Problem Analysis

Mathematical Concepts

Algebra
Radicals
Exponents

Formulas

\( \sqrt{a^n} = a^{n/2} \)
\( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \)

Theorems

Exponent Rules
Square Root Properties

Suitable Grade Level

Grades 9-11