Math Problem: Simplify $-36^{-rac{1}{2}}$

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Hey guys, let's tackle a cool math problem today that’s going to test your understanding of exponents and radicals. We’re diving deep into the world of negative exponents and fractional exponents to figure out which of the following is equivalent to -36^{-rac{1}{2}}? The options we’ve got are A. $-18$, B. $-6$, C. rac{1}{18}, and D. rac{1}{6}. This problem is a fantastic way to practice some fundamental mathematics principles, so grab your calculators, or better yet, let’s try to solve it without them!

Understanding Negative and Fractional Exponents

Before we jump into solving this specific problem, it’s super important to get a solid grasp on what negative and fractional exponents actually mean. Think of them as shortcuts or different ways of expressing operations we already know. Let’s break it down.

A negative exponent, like the $-1$ in our problem, tells us to take the reciprocal of the base. So, if you have something like $a^{-n}$, it’s the same as rac{1}{a^n}. The negative sign doesn't mean the whole result is negative; it just means we're flipping the number. For example, $2^{-3}$ is not $-8$, but rather rac{1}{2^3}, which equals rac{1}{8}. This is a crucial concept because it directly applies to the exponent in our question.

Now, let’s talk about fractional exponents. A fractional exponent, like rac{1}{2} in our case, represents a root. Specifically, a^{rac{1}{n}} is the $n$-th root of $a$. So, a^{rac{1}{2}} is the square root of $a$, a^{rac{1}{3}} is the cube root of $a$, and so on. When you see an exponent of rac{1}{2}, you should immediately think “square root.” For instance, 25^{rac{1}{2}} is the same as $\sqrt{25}$, which is $5$.

Combining these two rules is key to solving our problem. When you have a negative fractional exponent, like a^{-rac{1}{n}}, you first take the reciprocal because of the negative sign, giving you rac{1}{a^{rac{1}{n}}}. Then, you apply the fractional exponent rule, meaning rac{1}{a^{rac{1}{n}}} is equivalent to rac{1}{\sqrt[n]{a}}.

So, for our problem, we have -36^{-rac{1}{2}}. The base is $36$, the exponent is -rac{1}{2}. The negative sign tells us to take the reciprocal, and the rac{1}{2} tells us to take the square root. Let’s put these rules into action and see which of our answer choices it matches. This is where the real mathematics fun begins, guys!

Step-by-Step Solution

Alright, team, let’s get down to business and solve -36^{-rac{1}{2}}. We'll use the rules we just discussed to simplify this step-by-step. It’s all about breaking down the problem into manageable parts.

Step 1: Address the Negative Exponent.

Our expression is -36^{-rac{1}{2}}. The first thing we notice is the negative sign in the exponent. Remember our rule: a^{-n} = rac{1}{a^n}. Applying this to our problem, the base is $36$ and the exponent is rac{1}{2}. So, 36^{-rac{1}{2}} becomes rac{1}{36^{rac{1}{2}}}. The negative sign in the exponent flips the entire term. It’s important to note that the negative sign in front of the $36$ is not part of the base being raised to the power. The base is strictly $36$. If it were (-36)^{-rac{1}{2}}, that would be a different beast, potentially involving complex numbers. But here, it’s - (36^{-rac{1}{2}}).

So, we have transformed -36^{-rac{1}{2}} into - \frac{1}{36^{rac{1}{2}}}.

Step 2: Address the Fractional Exponent.

Now, let’s look at the term 36^{rac{1}{2}}. As we discussed, a fractional exponent of rac{1}{2} means we need to find the square root of the base. So, 36^{rac{1}{2}} is the same as $\sqrt{36}$.

What is the square root of $36$? We’re looking for a number that, when multiplied by itself, equals $36$. That number is $6$, because $6 imes 6 = 36$. So, $\sqrt{36} = 6$. Therefore, 36^{rac{1}{2}} = 6.

Step 3: Combine the Results.

We had simplified our expression in Step 1 to - \frac{1}{36^{rac{1}{2}}}. In Step 2, we found that 36^{rac{1}{2}} = 6. Now, we just substitute this value back into our expression:

- \frac{1}{36^{rac{1}{2}}} = - \frac{1}{6}.

So, the expression -36^{-rac{1}{2}} is equivalent to -rac{1}{6}.

Let's double-check our steps to make sure we haven’t missed anything. The original expression was -36^{-rac{1}{2}}.

1. The negative exponent (-rac{1}{2}) means we take the reciprocal of the base raised to the positive exponent: -(36^{-rac{1}{2}}) = -(\frac{1}{36^{rac{1}{2}}}).
2. The fractional exponent (rac{1}{2}) means we take the square root: 36^{rac{1}{2}} = \sqrt{36} = 6.
3. Substituting back: -(\frac{1}{6}) = -rac{1}{6}.

Our mathematics calculation is solid! The value equivalent to -36^{-rac{1}{2}} is indeed -rac{1}{6}.

Comparing with the Options

We’ve worked hard to simplify the expression -36^{-rac{1}{2}} and arrived at our answer: -rac{1}{6}. Now, it’s time to compare this with the given options to see which one is correct. Remember, we are looking for the option that is equivalent to our calculated value.

The options provided are:

A. $-18$ B. $-6$ C. rac{1}{18} D. rac{1}{6}

Let’s take a look:

• Option A: $-18$. Our result is -rac{1}{6}. These are clearly not the same. $-18$ is a large negative integer, while -rac{1}{6} is a small negative fraction.
• Option B: $-6$. Our result is -rac{1}{6}. Again, these are different. $-6$ is an integer, and -rac{1}{6} is a fraction. This option might tempt someone if they only considered the negative sign and the square root of 36, forgetting the reciprocal part of the negative exponent.
• Option C: rac{1}{18}. Our result is -rac{1}{6}. This option has the correct magnitude of the fraction but the wrong sign. The negative sign in the original expression is crucial!
• Option D: rac{1}{6}. Our result is -rac{1}{6}. This option has the correct magnitude of the fraction but the wrong sign. This is the positive counterpart to our answer.

Wait a minute, guys. It seems there might be a slight discrepancy between our calculated answer and the provided options. Let's re-examine the original problem statement and our solution very carefully.

The problem is: Which of the following is equivalent to -36^{-rac{1}{2}} ?

Our step-by-step simplification led us to -rac{1}{6}.

Let’s review the rules one last time:

1. Negative Exponent: a^{-n} = rac{1}{a^n}
2. Fractional Exponent: a^{rac{1}{n}} = \sqrt[n]{a}

Applying these to -36^{-rac{1}{2}}:

First, the negative exponent applies to the base $36$. So, 36^{-rac{1}{2}} = rac{1}{36^{rac{1}{2}}}.

Then, the fractional exponent rac{1}{2} means the square root: 36^{rac{1}{2}} = \sqrt{36} = 6.

So, 36^{-rac{1}{2}} = rac{1}{6}.

Now, consider the original expression: -36^{-rac{1}{2}}. The negative sign outside the base $36$ is separate from the exponent. This means we calculate 36^{-rac{1}{2}} first, and then apply the negative sign.

Therefore, -36^{-rac{1}{2}} = - (36^{-rac{1}{2}}) = - (\frac{1}{6}) = -rac{1}{6}.

My apologies, everyone! It appears I made a mistake in my previous comparison. Let me re-evaluate the options with the correct answer -rac{1}{6}.

Our calculated value is -rac{1}{6}.

Let's check the options again:

A. $-18$ B. $-6$ C. rac{1}{18} D. rac{1}{6}

None of the options directly match -rac{1}{6}. This suggests a potential typo in the question or the options provided. In a real test scenario, you would want to double-check the problem statement for any errors. However, if we assume the question intended to ask for something that is among the options, let’s think about possible interpretations or common mistakes.

Possible Interpretation 1: Misinterpreting the Negative Sign

If the question was written as (-36)^{-rac{1}{2}}, this would involve the square root of a negative number, leading to complex numbers, which are typically not covered in basic algebra. So, this interpretation is unlikely.

Possible Interpretation 2: A Typo in the Problem

• If the problem was 36^{rac{1}{2}}, the answer would be $6$ (Option B, but without the negative sign).
• If the problem was 36^{-rac{1}{2}}, the answer would be rac{1}{6} (Option D).
• If the problem was - rac{1}{36^{rac{1}{2}}}, the answer would be -rac{1}{6}.

Given the options, it's most probable that the question intended to be 36^{-rac{1}{2}}, in which case the answer would be rac{1}{6} (Option D).

However, strictly following the provided expression -36^{-rac{1}{2}}, the correct mathematical answer is -rac{1}{6}. Since this is not an option, let's reconsider the possibility of a typo in my own calculation or understanding.

Let's re-verify the base and exponent relationship. $a^{-n} = 1/a^n$. $a^{1/n} = ext{nth root of } a$.

So, $36^{-1/2} = 1 / (36^{1/2}) = 1 / (\sqrt{36}) = 1/6$.

The expression given is $-36^{-1/2}$. This means the negative sign is applied after the exponentiation.

So, $-36^{-1/2} = -(36^{-1/2}) = -(1/6) = -1/6$.

My calculations consistently lead to -rac{1}{6}. Let me look at the provided options one last time.

A. $-18$ B. $-6$ C. rac{1}{18} D. rac{1}{6}

It seems there is indeed an issue with the question or the options. However, in multiple-choice questions like this, sometimes you have to pick the