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Difficulty: MediumExponents, Roots, and Scientific Notation

What is the value of the expression 38×2662\frac{\sqrt{3^8 \times 2^6}}{6^2}?

  1. A
    11
  2. 18Answer
  3. C
    36
  4. D
    108
  5. E
    144

Answer

18
To evaluate the expression, first simplify the square root in the numerator. Since the square root of a product is the product of the square roots of its factors, we have 38×26=38/2×26/2=34×23\sqrt{3^8 \times 2^6} = 3^{8/2} \times 2^{6/2} = 3^4 \times 2^3. Next, rewrite the base in the denominator in terms of its prime factors: 62=(3×2)2=32×226^2 = (3 \times 2)^2 = 3^2 \times 2^2. Now, divide the simplified terms using the quotient rule for exponents: 34×2332×22=34−2×23−2=32×21\frac{3^4 \times 2^3}{3^2 \times 2^2} = 3^{4-2} \times 2^{3-2} = 3^2 \times 2^1. Evaluating this gives 9×2=189 \times 2 = 18.

Step-by-Step Solution

1
Simplify the square root in the numerator.
38×26=38×26=38/2×26/2=34×23\sqrt{3^8 \times 2^6} = \sqrt{3^8} \times \sqrt{2^6} = 3^{8/2} \times 2^{6/2} = 3^4 \times 2^3.
Applying the square root to factors with even exponents reduces their exponents by half.
2
Rewrite the denominator 626^2 in terms of prime bases 22 and 33.
62=(2×3)2=22×326^2 = (2 \times 3)^2 = 2^2 \times 3^2.
Expressing the bases in prime factors allows the use of exponent rules to simplify the fraction.
3
Divide the simplified numerator by the simplified denominator using the quotient rule.
34×2332×22=34−2×23−2=32×21=9×2=18\frac{3^4 \times 2^3}{3^2 \times 2^2} = 3^{4-2} \times 2^{3-2} = 3^2 \times 2^1 = 9 \times 2 = 18.
Subtracting exponents of like bases simplifies the rational expression to a final numerical value.

Key Concept

Simplifying radical expressions and applying laws of exponents for multiplication and division of powers.
Estimated Time:1m 0s
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Topics

Exponents, Roots, and Scientific Notation

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