Of course! Let's simplify the given expression [tex]\(\frac{30 x^6}{3 x^{-2}}\)[/tex] step by step.
### Step 1: Simplify the Coefficients
We start by simplifying the numerical coefficients separately from the variable parts. In the given expression, the coefficients are 30 in the numerator and 3 in the denominator.
[tex]\[
\frac{30}{3} = 10
\][/tex]
### Step 2: Simplify the Exponents
Next, handle the variable part [tex]\(x^6\)[/tex] and [tex]\(x^{-2}\)[/tex]. We use the properties of exponents for division:
[tex]\[
\frac{x^a}{x^b} = x^{a - b}
\][/tex]
Applying this property to our expression:
[tex]\[
\frac{x^6}{x^{-2}} = x^{6 - (-2)}
\][/tex]
### Step 3: Simplify the Exponent Calculation
Simplify the exponent by performing the subtraction inside the exponent:
[tex]\[
6 - (-2) = 6 + 2 = 8
\][/tex]
Thus,
[tex]\[
\frac{x^6}{x^{-2}} = x^8
\][/tex]
### Step 4: Combine the Coefficient and the Simplified Variable Part
Now we combine our simplified coefficient and the variable part:
[tex]\[
10 \times x^8
\][/tex]
### Final Simplified Expression
The simplified form of the given expression [tex]\(\frac{30 x^6}{3 x^{-2}}\)[/tex] is:
[tex]\[
10x^8
\][/tex]
Thus, the final answer is [tex]\(10 \times x^8\)[/tex].
