Simplifying Expressions with Exponents

Q1. Simplify the expression. Write the result using positive exponents only. Assume that all bases are not equal to 0.

$\frac{p^6 p}{p^{-1}}$

Background

Topic: Exponent Rules (Properties of Exponents)

This question tests your understanding of how to simplify expressions involving exponents, especially when multiplying and dividing powers with the same base. You are also asked to express your answer using only positive exponents.

Key Terms and Formulas:

• Product of Powers Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient of Powers Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Negative Exponent Rule: $a^{-n} = \frac{1}{a^n}$

Step-by-Step Guidance

1. First, combine the exponents in the numerator using the product of powers rule: $p^6 \cdot p = p^{6+1} = p^7$.

2. Rewrite the expression as $\frac{p^7}{p^{-1}}$.

3. Apply the quotient of powers rule: $\frac{p^7}{p^{-1}} = p^{7 - (-1)}$.

4. Simplify the exponent: $7 - (-1)$ becomes $7 + 1$.

Try solving on your own before revealing the answer!

Final Answer: $p^8$

By combining the exponents and applying the rules, the simplified expression with positive exponents is $p^8$.