Math Problem Statement

Which of the following is equivalent to (√32)(√⁵64)?

Solution

We are tasked with solving the following mathematical expression:

(32)(645)(\sqrt{32})(\sqrt[5]{64})

To begin, let's simplify both components of the product:

  1. Simplify 32\sqrt{32}: 32=16×2=16×2=42\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}

  2. Simplify 645\sqrt[5]{64}: [ \sqrt[5]{64} = \sqrt[5]{2^6} = 2^{6/5} ]

Now, we compute the product of these two simplified expressions:

(42)×(26/5)=4×21/2×26/5(4\sqrt{2}) \times (2^{6/5}) = 4 \times 2^{1/2} \times 2^{6/5}

Using the property of exponents, 2a×2b=2a+b2^{a} \times 2^{b} = 2^{a+b}:

4×21/2+6/5=4×2(5/10+12/10)=4×217/104 \times 2^{1/2 + 6/5} = 4 \times 2^{(5/10 + 12/10)} = 4 \times 2^{17/10}

Next, express 4 as 222^2:

22×217/10=22+17/10=2(20/10+17/10)=237/102^2 \times 2^{17/10} = 2^{2 + 17/10} = 2^{(20/10 + 17/10)} = 2^{37/10}

This matches with the answer choices. Since the form of the expression is 8×2k8 \times 2^{k} where k=7/5k = 7/5, this corresponds to option D:

8×210/27.\boxed{8 \times 2^{10/27}}.

Would you like further details or have any questions? Here are 5 related questions for practice:

  1. How do you simplify radical expressions involving prime factorization?
  2. What are the properties of exponents used to combine roots and powers?
  3. How do you simplify fractional exponents like 26/52^{6/5}?
  4. What is the significance of multiplying powers of the same base?
  5. Can you explain the process of converting between radicals and fractional exponents?

Tip: Always simplify radical expressions by first looking for perfect squares or cubes!

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Math Problem Analysis

Mathematical Concepts

Exponents and Roots
Radical Expressions
Simplification of Expressions

Formulas

√a × √b = √(a × b)
Exponent Rules: a^m × a^n = a^(m+n)
Conversion between Radicals and Fractional Exponents: n√(a^m) = a^(m/n)

Theorems

Exponent Properties
Radical Simplification

Suitable Grade Level

Grades 9-12