To simplify the expression [tex]\(\frac{20 v w^3}{36 v^4 w^3}\)[/tex], let's break it down step by step:
1. Simplify the coefficients (numerator and denominator):
The coefficient in the numerator is 20 and the coefficient in the denominator is 36. To simplify these coefficients, we can find the greatest common divisor (GCD) of the two numbers.
The GCD of 20 and 36 is 4. Thus, we divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{20}{4} = 5 \quad \text{and} \quad \frac{36}{4} = 9
\][/tex]
So the expression simplifies to:
[tex]\[
\frac{5 v w^3}{9 v^4 w^3}
\][/tex]
2. Simplify the variables with the same base (v and w):
- For [tex]\(v\)[/tex]:
In the numerator, [tex]\(v\)[/tex] is raised to the power of 1 (since [tex]\(v = v^1\)[/tex]).
In the denominator, [tex]\(v\)[/tex] is raised to the power of 4.
When we divide variables with the same base, we subtract the exponents:
[tex]\[
v^{1-4} = v^{-3}
\][/tex]
- For [tex]\(w\)[/tex]:
In both the numerator and the denominator, [tex]\(w\)[/tex] is raised to the power of 3. Since the exponent is the same, they cancel each other out:
[tex]\[
w^{3-3} = w^0 = 1 \quad \text{(anything raised to the power of 0 is 1)}
\][/tex]
3. Combine the simplified components:
After simplifying the coefficients and the variables, the expression reduces to:
[tex]\[
\frac{5 v^{-3} \cdot 1}{9 \cdot 1} = \frac{5 v^{-3}}{9}
\][/tex]
4. Rewriting with positive exponents (if desired):
The final result can be left as [tex]\(\frac{5 v^{-3}}{9}\)[/tex]. However, if we want to express it without negative exponents, we can rewrite it as:
[tex]\[
\frac{5}{9 v^3}
\][/tex]
So, the simplified form of the expression [tex]\(\frac{20 v w^3}{36 v^4 w^3}\)[/tex] is:
[tex]\[
\frac{5}{9 v^3}
\][/tex]
