Evaluate square root expression

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$$\sqrt{33^4 - 22^4 - 11^4} = ?$$

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Step by Step Written Solution

1
Step 1

Hi Türkan, let's solve this interesting radical expression together by using some clever factoring.

Solving the Expression

2
Step 2

Notice that thirty-three, twenty-two, and eleven are all multiples of eleven.

$$\sqrt{33^4 - 22^4 - 11^4}$$
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Step 3

We can rewrite each term using eleven as a factor: thirty-three is three times eleven, and twenty-two is two times eleven.

$$\sqrt{(3 \cdot 11)^4 - (2 \cdot 11)^4 - 11^4}$$
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Step 4

Now, distribute the exponent of four to each part inside the parentheses.

$$\sqrt{3^4 \cdot 11^4 - 2^4 \cdot 11^4 - 11^4}$$
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Step 5

Next, we factor out the common term, which is eleven to the power of four.

$$\sqrt{11^4 \cdot (3^4 - 2^4 - 1)}$$
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Step 6

Now, let's calculate the values inside the parentheses. Three to the power of four is eighty-one, two to the power of four is sixteen, and one to the power of four is simply one.

$$\sqrt{11^4 \cdot (81 - 16 - 1)}$$

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About This Question

Subject
Mathematics
Topic
Algebra
Difficulty
Medium
Question Type
Open Ended

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