Math  /  Algebra

QuestionSimplify (32)12(94)1273\frac{\left(3^{2}\right)^{-\frac{1}{2}}\left(9^{4}\right)^{-1}}{27^{-3}} a. 13-\frac{1}{3} c. 3 b. 131^{3} d. 1
Please select the best answer from the choices provided A B C D Mark this and retum Save and AssessmentViewer/Activit...

Studdy Solution

STEP 1

What is this asking? We need to simplify a big fraction with a bunch of exponents, and then pick the right answer. Watch out! Negative exponents can be tricky!
Remember, a negative exponent means "flip it," not "make it negative." Also, keep track of your bases – they all need to be the same to combine exponents.

STEP 2

1. Rewrite everything with the same base.
2. Simplify the numerator.
3. Simplify the fraction.

STEP 3

Let's **rewrite** everything using a base of 3!
We've got 323^2, 9=329 = 3^2, and 27=3327 = 3^3.
This is important because we can only combine exponents when the bases are the same.
It's like trying to add apples and oranges – you need to convert them to the same "fruit units" first!

STEP 4

So, our expression becomes: (32)12((32)4)1(33)3 \frac{\left(3^{2}\right)^{-\frac{1}{2}}\left(\left(3^{2}\right)^{4}\right)^{-1}}{\left(3^{3}\right)^{-3}}

STEP 5

Let's tackle the first term in the numerator: (32)12\left(3^{2}\right)^{-\frac{1}{2}}.
When you raise a power to a power, you **multiply** the exponents.
So, 212=12 \cdot -\frac{1}{2} = -1.
This gives us 313^{-1}.
Remember, a negative exponent means "flip it," so 31=133^{-1} = \frac{1}{3}.
It's like the exponent is saying, "I don't belong here!
I need to go to the other side of the fraction bar!"

STEP 6

Now for the second term in the numerator: ((32)4)1\left(\left(3^{2}\right)^{4}\right)^{-1}.
First, multiply the exponents inside the parentheses: 24=82 \cdot 4 = 8, giving us (38)1\left(3^{8}\right)^{-1}.
Now, multiply the exponents again: 81=88 \cdot -1 = -8, giving us 383^{-8}.
Again, that negative exponent wants to flip, so 38=1383^{-8} = \frac{1}{3^8}.

STEP 7

Putting the numerator together, we **multiply** the two terms: 3138=31+(8)=39=139 3^{-1} \cdot 3^{-8} = 3^{-1 + (-8)} = 3^{-9} = \frac{1}{3^9} Remember, when multiplying terms with the same base, we *add* the exponents.

STEP 8

Now, let's look at the denominator: (33)3\left(3^{3}\right)^{-3}.
Multiply the exponents: 33=93 \cdot -3 = -9, giving us 393^{-9}.
This becomes 139\frac{1}{3^9}.

STEP 9

Finally, we have: 139139 \frac{\frac{1}{3^9}}{\frac{1}{3^9}} To divide by a fraction, we **multiply** by its reciprocal.
The reciprocal of 139\frac{1}{3^9} is 391=39\frac{3^9}{1} = 3^9.
So, we have: 13939=3939=1 \frac{1}{3^9} \cdot 3^9 = \frac{3^9}{3^9} = 1 We're dividing 393^9 by itself, which always equals **1**!

STEP 10

Our simplified expression is **1**.
That matches answer choice *D*.

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