Solving √81 in Various Ways: A Comprehensive Guide

Mathematics is a fascinating field where problems can often be solved through multiple methods. One such intriguing example is solving the square root of 81, √81. Let's explore some of the ways to solve this problem and deepen our understanding of mathematical operations.

1. Traditional Square Root Calculation

The square root of a number is a value that, when multiplied by itself, gives the original number. For √81, the value is 9, because 9 * 9 81. This is the most straightforward and commonly used method for finding the square root of a perfect square like 81.

Let's verify this:

Verification

To confirm that √81 9:

$$sqrt{81} 9$$

Therefore, by squaring both sides:

$$(9) * (9) 81$$

This confirms that the square root of 81 is indeed 9.

2. Using the Square Law

The square law or the property of exponents states that ( a^n * a^m a^{n m} ). When solving for √81, we can use this property to simplify the expression:

$$sqrt{81} * sqrt{81} 81^1$$

Let's break this down step by step:

1. Write ( sqrt{81} ) as ( 81^{1/2} ).

$$ (81^{1/2}) * (81^{1/2}) 81^{1} $$

2. Using the property of exponents: (a^{1/2} * a^{1/2} a^{1/2 1/2} a^1 ).

$$ 81^{1/2 1/2} 81^1 $$

3. Simplify the exponent:

$$ 81^1 81 $$

4. Recall that (81 (9)^2), so:

$$ 81 9 * 9 $$

Hence, we have verified that (sqrt{81} 9).

3. Canceling Under the Root

Another method involves directly canceling out under the root. Since (sqrt{81} 9), we can write:

$$ sqrt{81} * sqrt{81} 81 $$

By canceling (sqrt{81} ) under the root, we simplify it to:

$$ 9 * 9 81 $$

Again, confirming that the square root of 81 is 9.

Conclusion

As seen, the square root of 81, √81, can be solved through various methods, all leading to the same result: (9). This multiplicity in solving not only reinforces our understanding of mathematical operations but also enhances problem-solving skills. Whether you use the traditional method, the square law, or cancellation techniques, the key is to ensure the correctness of the method and the final answer.

Key Takeaways

The square root of 81 is 9. Use the square law and properties of exponents to verify the solution. Understand that different methods can lead to the same correct result.

Related Keywords

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By exploring these methods, students and educators can gain a deeper insight into the world of mathematics and the beauty of its problem-solving techniques.