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Math

Math Snap

PROBLEM

What is the value of cc in the equation below?
5552=ab=c\frac{5^5}{5^2} = a^b = c

STEP 1

What is this asking?
We're asked to find the final value (cc) of an expression where we divide two exponents with the same base, then rewrite the result as a different power, and finally compute the numerical value.
Watch out!
Don't mix up the rules of exponents!
Remember, when dividing exponents with the same base, we subtract the powers, not divide them.

STEP 2

1. Simplify the fraction
2. Rewrite as a power of a different base
3. Calculate the final value

STEP 3

Alright, let's start with our awesome fraction: 5552\frac{5^5}{5^2}.
Remember the rule: when we divide exponents with the same base, we subtract the powers.
Why? Because we're essentially dividing out common factors.
It's like simplifying a fraction!
We're dividing both the top and bottom by 525^2.

STEP 4

So, we have 5552=5253521=531=53\frac{5^5}{5^2} = \frac{5^2 \cdot 5^3}{5^2 \cdot 1} = \frac{5^3}{1} = 5^3.
We're left with 55 raised to the power of 52=35-2=3, which is 535^3.

STEP 5

The problem says ab=53a^b = 5^3.
We need to find a value for cc where c=abc=a^b.
Notice something cool: the problem doesn't actually say that a and b have to be different from 5 and 3!
So let's just keep it simple.

STEP 6

We can just say a=5a=5 and b=3b=3.
That fits perfectly!
So, we have ab=53a^b = 5^3.

STEP 7

Now, we just need to calculate 535^3.
This means 55 multiplied by itself three times: 5555 \cdot 5 \cdot 5.

STEP 8

Let's do it step by step: 55=255 \cdot 5 = \textbf{25}.
Then, 255=12525 \cdot 5 = \textbf{125}.
So, 53=1255^3 = \textbf{125}.
Since c=ab=53c = a^b = 5^3, we have c=125c = \textbf{125}.

SOLUTION

Therefore, the value of cc is 125\textbf{125}.

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