Math / Algebra

QuestionWhat is the value of $c$ in the equation below?
$\frac{5^5}{5^2} = a^b = c$

Studdy Solution

STEP 1

What is this asking? We're asked to find the final value ( $c$ ) 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: $\frac{5^5}{5^2}$ .
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 $5^2$ .

STEP 4

So, we have $\frac{5^5}{5^2} = \frac{5^2 \cdot 5^3}{5^2 \cdot 1} = \frac{5^3}{1} = 5^3$ .
We're left with $5$ raised to the power of $5-2=3$ , which is $5^3$ .

STEP 5

The problem says $a^b = 5^3$ .
We need to find a value for $c$ where $c=a^b$ .
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 $a=5$ and $b=3$ .
That fits perfectly!
So, we have $a^b = 5^3$ .

STEP 7

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

STEP 8

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

STEP 9

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

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QuestionWhat is the value of $c$ in the equation below?
$\frac{5^5}{5^2} = a^b = c$

Studdy Solution

STEP 1

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: $\frac{5^5}{5^2}$ .
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 $5^2$ .

STEP 4

So, we have $\frac{5^5}{5^2} = \frac{5^2 \cdot 5^3}{5^2 \cdot 1} = \frac{5^3}{1} = 5^3$ .
We're left with $5$ raised to the power of $5-2=3$ , which is $5^3$ .

STEP 5

The problem says $a^b = 5^3$ .
We need to find a value for $c$ where $c=a^b$ .
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 $a=5$ and $b=3$ .
That fits perfectly!
So, we have $a^b = 5^3$ .

STEP 7

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

STEP 8

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

STEP 9

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

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QuestionWhat is the value of ccc in the equation below?5552=ab=c\frac{5^5}{5^2} = a^b = c5255​=ab=c

Studdy Solution

STEP 1

STEP 2

1. Simplify the fraction2. Rewrite as a power of a different base3. Calculate the final value

STEP 3

STEP 4

So, we have 5552=52⋅5352⋅1=531=53\frac{5^5}{5^2} = \frac{5^2 \cdot 5^3}{5^2 \cdot 1} = \frac{5^3}{1} = 5^35255​=52⋅152⋅53​=153​=53.We're left with 555 raised to the power of 5−2=35-2=35−2=3, which is 535^353.

STEP 5

The problem says ab=53a^b = 5^3ab=53.We need to find a value for ccc where c=abc=a^bc=ab.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=5a=5 and b=3b=3b=3.That fits perfectly!So, we have ab=53a^b = 5^3ab=53.

STEP 7

Now, we just need to **calculate** 535^353.This means 555 multiplied by itself three times: 5⋅5⋅55 \cdot 5 \cdot 55⋅5⋅5.

STEP 8

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

STEP 9

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

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QuestionWhat is the value of $c$ in the equation below?
$\frac{5^5}{5^2} = a^b = c$

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

So, we have $\frac{5^5}{5^2} = \frac{5^2 \cdot 5^3}{5^2 \cdot 1} = \frac{5^3}{1} = 5^3$ .
We're left with $5$ raised to the power of $5-2=3$ , which is $5^3$ .

The problem says $a^b = 5^3$ .
We need to find a value for $c$ where $c=a^b$ .
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.

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

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

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

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