(49x^2y)^1/2(27x^6y^3/2)^1/3

Jasonbradley

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Jun 16, 2024

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Simplifying the Expression (49x²y)^1/2(27x^6y^3/2)^1/3

This article will guide you through simplifying the expression (49x²y)^1/2(27x^6y^3/2)^1/3.

Understanding the Properties of Exponents

Before we begin, let's recall some important exponent properties:

• Product of Powers: (a^m) * (a^n) = a^(m+n)
• Power of a Product: (ab)^n = a^n * b^n
• Power of a Power: (a^m)^n = a^(m*n)
• Fractional Exponents: a^(1/n) = the nth root of a

Step-by-Step Simplification

1. Simplify each term individually:

   • (49x²y)^1/2: Applying the "power of a product" property: (49x²y)^1/2 = 49^(1/2) * x^(21/2) * y^(11/2) = 7x * y^(1/2)
   • (27x^6y^3/2)^1/3: Applying the "power of a product" property: (27x^6y^3/2)^1/3 = 27^(1/3) * x^(6*1/3) * y^(3/2 * 1/3) = 3x² * y^(1/2)

2. Combine the simplified terms: (7x * y^(1/2)) * (3x² * y^(1/2))

3. Apply the "product of powers" property: 7 * 3 * x^(1+2) * y^(1/2 + 1/2)

4. Simplify: 21x³y

Final Result

The simplified form of the expression (49x²y)^1/2(27x^6y^3/2)^1/3 is 21x³y.