Question

without evaluating, rewrite the expression using only positive exponents. (left( \frac{14^{-16} cdot 18^{13}}{14^7}
ight)^{-2}) use the number pad to enter your answer in the box.

Explanation:

Step1: Apply the negative exponent rule

The negative exponent rule states that \((\frac{a}{b})^{-n} = (\frac{b}{a})^{n}\). So for \((\frac{14^{-9} \cdot 18^{3}}{14^{7}})^{-2}\), we can rewrite it as \((\frac{14^{7}}{14^{-9} \cdot 18^{3}})^{2}\)

Step2: Apply the quotient rule for exponents

The quotient rule for exponents is \(\frac{a^{m}}{a^{n}} = a^{m - n}\). When we have \(\frac{14^{7}}{14^{-9}}\), using the quotient rule, we get \(14^{7 - (-9)}=14^{7 + 9}=14^{16}\)

Step3: Square the numerator and denominator

Now we have \((\frac{14^{16}}{18^{3}})^{2}\). Squaring the numerator and the denominator, we get \(\frac{(14^{16})^{2}}{(18^{3})^{2}}\)

Step4: Apply the power of a power rule

The power of a power rule is \((a^{m})^{n}=a^{m\times n}\). So \((14^{16})^{2} = 14^{16\times2}=14^{32}\) and \((18^{3})^{2}=18^{3\times2}=18^{6}\)

Answer:

\(\frac{14^{32}}{18^{6}}\)