91Ó°ÊÓ

Understanding exponent rules is essential for simplifying algebraic expressions, especially when dealing with polynomial multiplication and exponentiation. Exponents, often referred to as powers, indicate how many times a number (the base) is multiplied by itself. For instance, in the expression \( 2^{3} \), 2 is the base and 3 is the exponent, meaning 2 is multiplied by itself three times (2 * 2 * 2 = 8). Key rules include:
\(\text{Product Rule: } a^{m} \times a^{n} = a^{m+n} \)
\(\text{Quotient Rule: } \frac{a^{m}}{a^{n}} = a^{m-n} \)
\(\text{Power Rule: } (a^{m})^{n} = a^{m \times n} \)
For example, applying the power rule in our exercise, we simplify \( (q^{4})^{2} \) as follows: \( q^{4 \times 2} = q^{8} \). Understanding these rules allows you to simplify and handle complex algebraic expressions efficiently.

When simplifying expressions involving fractions raised to an exponent, each part of the fraction is raised to the power separately. This operation is a direct application of the exponent rules. For instance, in the exercise, we have \( \frac{7}{9} \) raised to the power of 2:
\(\text{Step 1:} \) \((\frac{7}{9})^{2} \) means \( \frac{7^{2}}{9^{2}} \)
\( \text{Step 2: Simplify terms} \)
\(7^{2} = 49 \)
\(9^{2} = 81 \)
Bringing it all together, we get \( \frac{49}{81} \).
Fraction exponentiation simplifies complex fractions and allows easier manipulation of algebraic fractions.

Polynomial multiplication involves multiplying each term in the first polynomial by each term in the second polynomial. When multiplying monomials with coefficients and variables, multiply the coefficients together and then apply the product rule to the variables. In the exercise, we've simplified each term separately, and the multiplication principles apply similarly:

• For coefficients: \( \frac{49}{81}\)
• For variables: \( p^{2}\) and \( q^{8}\)

Combining terms after exponentiation:

• \text{Final answer: \( \frac{49}{81}p^{2}q^{8}\)

Hence, polynomial multiplication, combined with exponent rules, allows streamlined expression simplification.