Simplify (p^8)/(p^3)

Solution:

Simplification Process:

Step 1: Break down $p^8$ by separating $p^3$ from it. Write it as $\frac{p^3\cdot p^5}{p^3}$.

Step 2: Eliminate identical factors from numerator and denominator.

Step 2.1: Introduce multiplication by $1$ to clarify the process: $\frac{p^3\cdot p^5}{p^3\cdot 1}$.

Step 2.2: Strike out the matching factors: $\frac{\cancel{p^3}\cdot p^5}{\cancel{p^3}\cdot 1}$.

Step 2.3: Reformulate the fraction: $\frac{p^5}{1}$.

Step 2.4: Compute the division of $p^5$ by $1$: $p^5$.

Knowledge Notes:

When simplifying expressions with exponents, we use the laws of exponents. Here are the relevant points:

1. Multiplication of Like Bases: When multiplying powers with the same base, you add the exponents. For example, $a^m\cdot a^n=a^{m+n}$.

2. Division of Like Bases: When dividing powers with the same base, you subtract the exponents. For example, $\frac{a^m}{a^n}=a^{m-n}$, provided $a\ne 0$.

3. Exponent of One: Any number raised to the power of one is the number itself, e.g., $a^1=a$.

4. Exponent of Zero: Any non-zero number raised to the power of zero is one, e.g., $a^0=1$.

5. Cancellation Law: When a factor appears in both the numerator and the denominator of a fraction, it can be canceled out. This is because any number divided by itself equals one.

In the given problem, we apply the division law of exponents to simplify the expression $\frac{p^8}{p^3}$ by subtracting the exponents. We get $p^{8-3}=p^5$. Since dividing by 1 does not change the value of a number, $\frac{p^5}{1}$ simplifies to $p^5$.