Simplify $\sqrt{0.08^{12}}$: A Math Solution

When faced with mathematical expressions, particularly those involving exponents and roots, the goal is often to simplify them into a more manageable form. Let's tackle the expression $\sqrt{0.08^{12}}$. This problem tests your understanding of exponent rules and how they interact with square roots. We'll break down the process step-by-step to arrive at the correct answer, making sure to explain the reasoning behind each transformation. By the end of this discussion, you'll be confident in simplifying similar expressions involving roots and powers.

Understanding the Properties of Exponents and Roots

Before we dive into solving $\sqrt{0.08^{12}}$, it's crucial to recall some fundamental properties of exponents and roots. One of the most important properties for this problem is that a square root can be expressed as an exponent of $\frac{1}{2}$. In other words, $\sqrt{a} = a^{\frac{1}{2}}$. Another key property is the "power of a power" rule, which states that $(a^m)^n = a^{m \times n}$. Applying these rules systematically will allow us to simplify the given expression efficiently and accurately.

To simplify $\sqrt{0.08^{12}}$, we can first rewrite the square root as an exponent. As mentioned, $\sqrt{x}$ is equivalent to $x^{\frac{1}{2}}$. Therefore, $\sqrt{0.08^{12}}$ can be rewritten as $(0.08^{12})^{\frac{1}{2}}$. Now, we can apply the power of a power rule. In this case, $a = 0.08$, $m = 12$, and $n = \frac{1}{2}$. According to the rule $(a^m)^n = a^{m \times n}$, we multiply the exponents: $12 \times \frac{1}{2}$. This multiplication results in $6$. So, the expression simplifies to $0.08^6$. This matches option A among the choices provided.

Let's elaborate on why this simplification works. The square root operation essentially asks, "What number, when multiplied by itself, gives us the number inside the root?" When we have $0.08^{12}$, we are looking for a number that, when squared, equals $0.08^{12}$. If we consider $0.08^6$, and we square it, we get $(0.08^6)^2$. Using the power of a power rule, this becomes $0.08^{6 \times 2}$, which is $0.08^{12}$. Thus, $0.08^6$ is indeed the square root of $0.08^{12}$. This confirms our result. It is important to distinguish between the base number and the exponent. Here, the base is $0.08$, and its exponent is being manipulated. The choices provided offer different bases and exponents, so it's essential to follow the rules precisely to avoid common mistakes. For instance, one might incorrectly assume that the square root applies directly to the base $0.08$ or that the exponent $12$ remains unchanged. However, the rules of exponents are clear: when raising a power to another power, the exponents are multiplied.

Step-by-Step Solution

Let's walk through the simplification of $\sqrt{0.08^{12}}$ step-by-step to ensure clarity and accuracy. We start with the given expression: $\sqrt{0.08^{12}}$.

Step 1: Rewrite the square root as an exponent. The square root of a number is the same as raising that number to the power of $\frac{1}{2}$. So, we can rewrite the expression as:

$$(0.08^{12})^{\frac{1}{2}}$$

This step is fundamental because it allows us to use the rules of exponents, which are often easier to manipulate than root symbols.

Step 2: Apply the "power of a power" rule. The "power of a power" rule states that when you raise a power to another power, you multiply the exponents: $(a^m)^n = a^{m \times n}$. In our expression, $a = 0.08$, $m = 12$, and $n = \frac{1}{2}$. Applying this rule, we get:

$$0.08^{(12 \times \frac{1}{2})}$$

Step 3: Calculate the new exponent. Now, we simply perform the multiplication of the exponents:

$$12 \times \frac{1}{2} = \frac{12}{2} = 6$$

Step 4: Write the final simplified expression. Substituting the calculated exponent back into the expression, we get:

$$0.08^6$$

This simplified form directly corresponds to option A. It's a straightforward application of exponent rules, making complex-looking expressions much easier to handle.

Analyzing the Options

Let's briefly examine why the other options are incorrect. Understanding why wrong answers are wrong can reinforce your grasp of the concept.

• A. $(0.08)^6$: As we've shown, this is the correct simplification.
• B. $(0.8)^6$: This option changes the base from $0.08$ to $0.8$. The square root operation does not alter the base itself, only the exponent of the number inside the root. Therefore, this is incorrect.
• C. $(0.16)^6$: This option not only changes the exponent calculation but also incorrectly modifies the base. It might arise from a misunderstanding of how the square root affects the number inside.
• D. $(0.12)^8$: This option seems to have confused the operation. Perhaps there was an attempt to divide the exponent by the root index (which is 2 for a square root) and then incorrectly apply the original exponent. However, the base remains $0.08$, and the exponent calculation is $12 \div 2 = 6$. This option is incorrect.

Conclusion

Simplifying $\sqrt{0.08^{12}}$ is a clear demonstration of how understanding and applying the fundamental rules of exponents can make complex mathematical expressions manageable. By rewriting the square root as an exponent of $\frac{1}{2}$ and then using the "power of a power" rule, we found that $\sqrt{0.08^{12}}$ simplifies to $0.08^6$. This straightforward process yields option A as the correct answer. Remember, when you encounter similar problems, always look for opportunities to apply these core mathematical properties. Consistent practice with these rules will build your confidence and proficiency in algebra.

For further exploration into the fascinating world of algebraic simplification and exponent rules, you can refer to resources like Khan Academy. They offer comprehensive explanations and practice problems that can deepen your understanding of these essential mathematical concepts.