To simplify the expression [tex]\(\frac{25 \times 5^2 \times x^8}{10^3 \times x^5}\)[/tex], we can follow a step-by-step process to simplify both the numerical and variable components.
### Step 1: Simplify the Numerical Component
First, let's simplify the numerical part of the fraction:
[tex]\[ 25 \times 5^2 \][/tex] is equal to [tex]\( 25 \times 25 \)[/tex], because [tex]\( 5^2 = 25 \)[/tex].
[tex]\[ 25 \times 25 = 625 \][/tex]
So, the numerator becomes:
[tex]\[ 625 \][/tex]
Now, let's look at the denominator:
[tex]\[ 10^3 \][/tex]
[tex]\[ 10^3 = 1000 \][/tex]
### Step 2: Simplify the Variable Component
Now, let's simplify the variable part of the fraction:
[tex]\[ \frac{x^8}{x^5} \][/tex]
Using the property of exponents that states [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex], we get:
[tex]\[ \frac{x^8}{x^5} = x^{8-5} = x^3 \][/tex]
### Step 3: Combine Both Components
Now that we have simplified both parts, we can put them together:
[tex]\[ \frac{625 \times x^8}{1000 \times x^5} = \frac{625}{1000} \times \frac{x^8}{x^5} \][/tex]
[tex]\[ = \frac{625}{1000} \times x^3 \][/tex]
Next, we simplify the fraction [tex]\(\frac{625}{1000}\)[/tex].
### Step 4: Simplify the Fraction
To simplify [tex]\(\frac{625}{1000}\)[/tex], we find the greatest common divisor (GCD) of 625 and 1000. The GCD is 125.
[tex]\[ \frac{625}{1000} = \frac{625 \div 125}{1000 \div 125} = \frac{5}{8} \][/tex]
### Step 5: Write the Simplified Expression
Now, combining the simplified numerical and variable parts, we get:
[tex]\[ \frac{5}{8} \times x^3 \][/tex]
Therefore, the simplified form of the expression is:
[tex]\[
\frac{25 \times 5^2 \times x^8}{10^3 \times x^5} = \frac{5}{8} x^3
\][/tex]
