Solve Nested Radicals: Cube Root of Square Root of 16 × Square Root Problems

Question

Complete the following exercise:

16316= \sqrt[3]{\sqrt{16}}\cdot\sqrt[]{\sqrt{16}}=

Video Solution

Solution Steps

00:00 Solve the following problem
00:03 A 'regular' root raised to the second power
00:14 When there is a root of order (C) to root (B)
00:17 The result equals the root of the orders' product
00:22 Apply this formula to our exercise
00:31 Let's calculate the product of the orders
00:39 When we have a root of order (C) on number (A) to the power of (B)
00:43 The result equals number (A) to the power of (B divided by C)
00:47 Apply this formula to our exercise, each number is to the power of 1
01:00 When we have a multiplication between powers with equal bases
01:03 The result equals the base with the power equal to the sum of the powers
01:08 Apply this formula to our exercise
01:13 Let's find a common denominator and connect between the powers
01:19 This is the solution

Step-by-Step Solution

To solve this problem, we will simplify the expression 16316 \sqrt[3]{\sqrt{16}} \cdot \sqrt[]{\sqrt{16}} using the rules for exponents and roots.

First, consider the inner square root 16 \sqrt{16} . We know that:
16=161/2 \sqrt{16} = 16^{1/2}

Next, we address the cube root term 163 \sqrt[3]{\sqrt{16}} . Express 16\sqrt{16} as 161/216^{1/2}, then:

  • 163=161/23\sqrt[3]{\sqrt{16}} = \sqrt[3]{16^{1/2}} Convert to exponents: 161/23=(161/2)1/3=16(1/2)(1/3)=161/6 \sqrt[3]{16^{1/2}} = (16^{1/2})^{1/3} = 16^{(1/2) \cdot (1/3)} = 16^{1/6}
  • The other term is 16=161/2\sqrt{\sqrt{16}} = \sqrt{16^{1/2}} Convert to exponents: 161/2=(161/2)1/2=16(1/2)(1/2)=161/4 \sqrt{16^{1/2}} = (16^{1/2})^{1/2} = 16^{(1/2) \cdot (1/2)} = 16^{1/4}

Now, multiply these results:
161/6161/4 16^{1/6} \cdot 16^{1/4}

Using the product rule for exponents (aman=am+n)(a^m \cdot a^n = a^{m+n}), combine the exponents:
161/6+1/4 16^{1/6 + 1/4}

Find the common denominator to add the fractions:

  • 1/6=2/121/6 = 2/12
  • 1/4=3/121/4 = 3/12 1/6+1/4=2/12+3/12=5/12 1/6 + 1/4 = 2/12 + 3/12 = 5/12

Thus, the expression becomes:
165/12 16^{5/12}

Therefore, the simplified expression is 16512 16^{\frac{5}{12}} .

Answer

16512 16^{\frac{5}{12}}