Nested square roots can look intimidating. But there is a powerful pattern: every square-root symbol divides the exponent by 2. In the video, four nested square roots turn an exponent of 48 into 3, giving the final answer 2³ = 8.
This shortcut is useful for algebra, exponents, radical expressions, homework, and test preparation. Once you learn how to count the square-root symbols, the problem becomes a simple division exercise.
The Expression in the Video
The video shows four square roots nested inside one another, with the entire expression raised to the 48th power:
The important detail is the number of nested square roots. There are four radical signs, so the exponent must be divided by 2 four times.
The Quick Answer
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The answer is 8. The trick is to reduce the exponent step by step.
Step-by-Step: Divide the Exponent by 2
Step 1: Start with the exponent
The expression has an exponent of 48:
Step 2: Count the nested square roots
There are four square-root signs. Each one divides the exponent by 2:
Step 3: Simplify one division at a time
Step 4: Evaluate the remaining power
After the four divisions, the exponent is 3:
Why Does Every Square Root Divide the Exponent?
A square root is the same as raising a number to the power of one-half:
When a power is raised to another power, multiply the exponents. Therefore:
With more than one nested square root, the exponent is divided by 2 repeatedly. Four nested square roots produce:
So the expression can be rewritten as:
The General Rule
If a positive number a has k nested square roots and the entire expression is raised to the power n, use this rule:
In simple words:
More Examples
Example 1: Two nested square roots
Consider:
There are two square roots, so divide 8 by 2 twice:
Example 2: Three nested square roots
Now consider:
There are three square roots:
Example 3: Four nested square roots
For the same four-root pattern with a different exponent:
Divide 32 by 2 four times:
Common Mistakes to Avoid
The most common mistake is dividing the exponent by 2 only once. If the expression has four nested square roots, divide by 2 four times. For an exponent of 48, stopping after one division gives 24, but the correct reduced exponent is 3.
Another mistake is counting parentheses instead of radical signs. Count the actual square-root symbols. Each radical sign represents one division by 2.
Do not change the base incorrectly. The base remains 2 in the video’s expression. Only the exponent changes as the nested roots are simplified.
Finally, keep the parentheses clear. The exponent 48 applies to the entire group of nested roots, not only to the number 2.
Practice Problems
Try these problems without a calculator:
- [√(√2)]⁸
- [√(√(√3))]¹⁶
- [√(√(√(√5)))]¹⁶
- [√(√(√7))]²⁴
- [√(√(√(√2)))]⁶⁴
Answer Key
- [√(√2)]⁸ = 2² = 4
- [√(√(√3))]¹⁶ = 3² = 9
- [√(√(√(√5)))]¹⁶ = 5
- [√(√(√7))]²⁴ = 7³ = 343
- [√(√(√(√2)))]⁶⁴ = 2⁴ = 16
Final Takeaway
Nested square roots are easier when you treat every radical as an exponent divider. For the expression in the video, there are four square roots:
Then simplify the remaining power:
So the complete answer is:
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