Simplifying Exponential Expressions: (6^(1/2))^2

Hey guys! Let's break down this math problem together. We're going to figure out the value of the expression ${\left(6^{1 / 2}\right)^2}$. This involves understanding exponents and how they work, especially when dealing with fractional exponents. Don't worry, it's not as intimidating as it looks! By the end of this, you'll be a pro at simplifying expressions like this.

Understanding the Basics

Before we dive into the problem, let's quickly review some exponent rules. Remember that an exponent tells you how many times to multiply a number by itself. For example, ${2^3}$ means ${2 \times 2 \times 2 = 8}$. A fractional exponent, like ${\frac{1}{2}}$, represents a root. Specifically, ${x^{1/2}}$ is the same as ${\sqrt{x}}$, which means the square root of x. This is a key concept for solving our problem. Also, remember the power of a power rule: ${(a^m)^n = a^{m \times n}}$. This rule tells us that when we raise a power to another power, we multiply the exponents.

Breaking Down the Expression

Now, let's look at our expression: ${\left(6^{1 / 2}\right)^2}$. The first thing to recognize is that ${6^{1/2}}$ is the same as ${\sqrt{6}}$. So, we can rewrite the expression as ${\left(\sqrt{6}\right)^2}$. Now, we need to square the square root of 6. Squaring a square root essentially cancels out the root. Think of it this way: if you take the square root of a number and then square it, you end up with the original number. Mathematically, ${\left(\sqrt{x}\right)^2 = x}$. Applying this to our problem, ${\left(\sqrt{6}\right)^2 = 6}$.

Applying the Power of a Power Rule

Alternatively, we can use the power of a power rule directly. Our expression is ${\left(6^{1 / 2}\right)^2}$. According to the rule ${(a^m)^n = a^{m \times n}}$, we multiply the exponents: ${\frac{1}{2} \times 2 = 1}$. So, our expression becomes ${6^1}$, which is simply 6. Both methods lead us to the same answer, which is 6.

Why the Other Options Are Incorrect

Let's quickly look at why the other options are wrong:

• B. 12: This is incorrect because it seems like someone might have multiplied 6 by 2 instead of understanding the exponent. There's no multiplication of 6 by 2 happening here.
• C. ${\sqrt{6}}$: This is the square root of 6, which is only the first part of the expression. We need to square it, so this isn't the final answer.
• D. ${2 \sqrt{6}}$: This is also incorrect. It looks like someone might have tried to combine the square root with some multiplication, but that's not what the expression dictates.

Conclusion

So, the correct answer is A. 6. We found this by recognizing that ${6^{1/2}}$ is the square root of 6, and squaring it cancels out the square root, leaving us with 6. Alternatively, we used the power of a power rule to multiply the exponents, which also gave us 6. Keep practicing these exponent rules, and you'll become a math whiz in no time! Remember, math is all about understanding the rules and applying them correctly.

Let's try another example to solidify your understanding. What if we had the expression ${\left(9^{1 / 2}\right)^2}$? Can you solve it? Think about what ${9^{1/2}}$ means and how squaring it affects the result. You got this!

Practice Problems for You

To ensure you've truly grasped the concept, let's work through a few more examples. These practice problems will help reinforce your understanding of fractional exponents and the power of a power rule. Remember, the key is to break down each problem step by step and apply the rules we've discussed.

Practice Problem 1: Evaluate ${\left(4^{1 / 2}\right)^2}$

Solution:

First, recognize that ${4^{1/2}}$ is the same as ${\sqrt{4}}$. The square root of 4 is 2. So, we have ${\left(2\right)^2}$, which equals 4. Alternatively, using the power of a power rule, we have ${\left(4^{1 / 2}\right)^2 = 4^{(1/2) * 2} = 4^1 = 4}$. Therefore, the value of the expression is 4.

Practice Problem 2: Simplify ${\left(16^{1 / 2}\right)^2}$

Solution:

Here, ${16^{1/2}}$ is the same as ${\sqrt{16}}$. The square root of 16 is 4. Thus, we have ${\left(4\right)^2}$, which equals 16. Using the power of a power rule, ${\left(16^{1 / 2}\right)^2 = 16^{(1/2) * 2} = 16^1 = 16}$. So, the simplified expression is 16.

Practice Problem 3: What is the value of ${\left(25^{1 / 2}\right)^2}$?

Solution:

We know that ${25^{1/2}}$ is the same as ${\sqrt{25}}$. The square root of 25 is 5. Therefore, we have ${\left(5\right)^2}$, which equals 25. Applying the power of a power rule, ${\left(25^{1 / 2}\right)^2 = 25^{(1/2) * 2} = 25^1 = 25}$. The value of the expression is 25.

Practice Problem 4: Calculate ${\left(36^{1 / 2}\right)^2}$

Solution:

In this case, ${36^{1/2}}$ is the same as ${\sqrt{36}}$. The square root of 36 is 6. Thus, we have ${\left(6\right)^2}$, which equals 36. Using the power of a power rule, ${\left(36^{1 / 2}\right)^2 = 36^{(1/2) * 2} = 36^1 = 36}$. The calculated value is 36.

Practice Problem 5: Evaluate ${\left(49^{1 / 2}\right)^2}$

Solution:

Here, ${49^{1/2}}$ is the same as ${\sqrt{49}}$. The square root of 49 is 7. So, we have ${\left(7\right)^2}$, which equals 49. Alternatively, using the power of a power rule, we have ${\left(49^{1 / 2}\right)^2 = 49^{(1/2) * 2} = 49^1 = 49}$. Therefore, the value of the expression is 49.

Advanced Tips

• Remember the Power of a Power Rule: Always keep in mind that ${(a^m)^n = a^{m \times n}}$. This rule is fundamental for simplifying exponential expressions.
• Fractional Exponents and Roots: Understand that ${x^{1/2}}$ is equivalent to ${\sqrt{x}}$, ${x^{1/3}}$ is equivalent to ${\sqrt[3]{x}}$, and so on. This understanding will make simplifying expressions much easier.
• Negative Exponents: Keep in mind that a negative exponent means taking the reciprocal of the base raised to the positive exponent. For example, ${a^{-n} = \frac{1}{a^n}}$.
• Combining Exponents: When multiplying numbers with the same base, add the exponents: ${a^m \times a^n = a^{m+n}}$. When dividing numbers with the same base, subtract the exponents: ${\frac{a^m}{a^n} = a^{m-n}}$.

Real-World Applications

Understanding exponents and roots isn't just for math class. They have real-world applications in various fields:

• Computer Science: Exponents are used extensively in computer science, especially in algorithms and data structures. For example, binary search algorithms use logarithmic time complexity, which involves exponents.
• Finance: Compound interest calculations involve exponents. The formula for compound interest is ${A = P(1 + r/n)^{nt}}$, where A is the final amount, P is the principal, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years.
• Physics: Exponents are used in physics to describe various phenomena, such as the inverse square law for gravitational force and electric force.
• Engineering: Engineers use exponents in many calculations, such as determining the strength of materials and designing structures.

Common Mistakes to Avoid

• Forgetting the Power of a Power Rule: One common mistake is forgetting to multiply the exponents when raising a power to another power. Always remember that ${(a^m)^n = a^{m \times n}}$.
• Misunderstanding Fractional Exponents: Make sure to understand that a fractional exponent represents a root. For example, ${x^{1/2}}$ is the square root of x, not half of x.
• Incorrectly Applying Negative Exponents: Remember that a negative exponent means taking the reciprocal, not making the base negative.
• Order of Operations: Always follow the correct order of operations (PEMDAS/BODMAS) when simplifying expressions.

Final Thoughts

Mastering exponents and roots is a fundamental skill in mathematics. By understanding the basic rules and practicing regularly, you'll be able to simplify complex expressions with ease. Remember to break down each problem step by step and apply the rules correctly. Keep practicing, and you'll become a math pro in no time!