Write the Fraction in Simplest Form ((-2s^3)^4)/(5s^-1)

Solution:

Step 1:

Rewrite the denominator's negative exponent as a positive exponent in the numerator by using the rule $b^{-n} = \frac{1}{b^n}$.

$$\frac{(-2s^3)^4 \cdot s}{5}$$

Step 2:

Begin simplifying the expression in the numerator.

Step 2.1:

Apply the power of a product rule to $(-2s^3)$.

$$\frac{(-2)^4 (s^3)^4 \cdot s}{5}$$

Step 2.2:

Calculate $(-2)^4$.

$$\frac{16 (s^3)^4 \cdot s}{5}$$

Step 2.3:

Evaluate the power of a power expression $(s^3)^4$.

Step 2.3.1:

Use the rule $(a^m)^n = a^{mn}$ to simplify the expression.

$$\frac{16 s^{3 \cdot 4} \cdot s}{5}$$

Step 2.3.2:

Perform the multiplication of the exponents $3$ and $4$.

$$\frac{16 s^{12} \cdot s}{5}$$

Step 2.4:

Combine the exponents of $s$.

Step 2.4.1:

Express $s$ as $s^1$.

$$\frac{16 s^1 \cdot s^{12}}{5}$$

Step 2.4.2:

Combine the exponents using the rule $a^m \cdot a^n = a^{m+n}$.

$$\frac{16 s^{1 + 12}}{5}$$

Step 2.4.3:

Add the exponents $1$ and $12$.

$$\frac{16 s^{13}}{5}$$

Knowledge Notes:

To simplify the given expression, we use several rules of exponents:

1. Negative Exponent Rule: $b^{-n} = \frac{1}{b^n}$, which allows us to move a term with a negative exponent from the denominator to the numerator or vice versa by changing the sign of the exponent.

2. Power of a Product Rule: $(ab)^n = a^n b^n$, which states that when two or more bases are multiplied and raised to a power, we can apply the power to each base individually.

3. Power of a Power Rule: $(a^m)^n = a^{mn}$, which indicates that when an exponent is raised to another exponent, we multiply the exponents.

4. Product of Powers Rule: $a^m \cdot a^n = a^{m+n}$, which tells us that when multiplying like bases, we add the exponents.

By applying these rules systematically, we can simplify the expression to its simplest form. In this case, we also need to perform basic arithmetic operations such as raising a number to a power and adding exponents.