Question

select all the expressions that are equivalent to \\(\frac{3^{18}}{3^{-2}}\\).
\\(3^{20}\\) \\(3^{16}\\) \\(\frac{1}{3^{-9}}\\) \\(\frac{1}{3^{-20}}\\)

Explanation:

Step1: Recall the exponent rule for division

When dividing two numbers with the same base, we subtract the exponents: \( \frac{a^m}{a^n}=a^{m - n} \). For \( \frac{3^{18}}{3^{-2}} \), here \( a = 3 \), \( m=18 \), \( n=- 2 \). So we calculate \( 18-(-2)=18 + 2=20 \). Thus \( \frac{3^{18}}{3^{-2}}=3^{20} \).

Step2: Recall the negative exponent rule

The negative exponent rule states that \( \frac{1}{a^{-n}}=a^{n} \). So for \( \frac{1}{3^{-20}} \), using the rule, we get \( 3^{20} \), which is equivalent to \( \frac{3^{18}}{3^{-2}} \).

Now let's check the other options:

• For \( 3^{16} \): \( 18-(-2)=20

eq16 \), so it's not equivalent.

• For \( \frac{1}{3^{-9}} \): Using the negative exponent rule, \( \frac{1}{3^{-9}} = 3^{9}

eq3^{20} \), so it's not equivalent.

Answer:

\( 3^{20} \), \( \frac{1}{3^{-20}} \)