Question

which expressions are equivalent to \\(\frac{3^{20}}{3^{2} cdot 3^{8}}\\)? select all that apply. \\(\frac{3^{20}}{3^{10}}\\) \\(\frac{3^{20}}{3^{16}}\\) \\(3^{20 - 10}\\) \\(3^{20 - 16}\\) \\(3^{4}\\) \\(3^{10}\\)

Explanation:

To determine which expressions are equivalent to \(\frac{3^{20}}{3^{2}\cdot3^{8}}\), we use the properties of exponents. Recall that when multiplying exponents with the same base, we add the exponents: \(a^m \cdot a^n = a^{m + n}\). And when dividing exponents with the same base, we subtract the exponents: \(\frac{a^m}{a^n}=a^{m - n}\).

Step 1: Simplify the denominator

First, simplify the denominator \(3^{2}\cdot3^{8}\) using the product rule of exponents (\(a^m \cdot a^n = a^{m + n}\)):
\[
3^{2}\cdot3^{8}=3^{2 + 8}=3^{10}
\]
So the original expression \(\frac{3^{20}}{3^{2}\cdot3^{8}}\) becomes \(\frac{3^{20}}{3^{10}}\).

Step 2: Simplify the fraction

Now, simplify \(\frac{3^{20}}{3^{10}}\) using the quotient rule of exponents (\(\frac{a^m}{a^n}=a^{m - n}\)):
\[
\frac{3^{20}}{3^{10}} = 3^{20-10}=3^{10}
\]
We can also rewrite \(\frac{3^{20}}{3^{10}}\) as is, or express it as \(3^{20 - 10}\) (which simplifies to \(3^{10}\)) or \(3^{20-16}\) is incorrect (since \(20 - 16 = 4\), and \(3^4\) is not equivalent). Let's check each option:

• \(\frac{3^{20}}{3^{10}}\): This is equivalent as we saw from Step 1.
• \(\frac{3^{20}}{3^{16}}\): Wait, no, the second option is \(\frac{3^{20}}{3^{16}}\)? Wait, the options are \(\frac{3^{20}}{3^{10}}\), \(\frac{3^{20}}{3^{16}}\), \(3^{20 - 10}\), \(3^{20 - 16}\), \(3^4\), \(3^{10}\). Wait, let's re - evaluate:

Wait, the denominator was \(3^{2}\cdot3^{8}=3^{10}\), so:

• \(\frac{3^{20}}{3^{10}}\): Equivalent (same as the simplified form of the original expression before applying the quotient rule).
• \(\frac{3^{20}}{3^{16}}\): Let's see, \(3^{16}\) is not equal to \(3^{10}\), so \(\frac{3^{20}}{3^{16}}=3^{20 - 16}=3^{4}\), which is not equivalent to the original expression.
• \(3^{20 - 10}\): Since \(3^{20-10}=3^{10}\), which is equivalent.
• \(3^{20 - 16}\): \(3^{20-16}=3^{4}\), not equivalent.
• \(3^4\): Not equivalent.
• \(3^{10}\): Equivalent (since \(3^{20-10}=3^{10}\)).

So the equivalent expressions are \(\frac{3^{20}}{3^{10}}\), \(3^{20 - 10}\), and \(3^{10}\).

Answer:

\(\frac{3^{20}}{3^{10}}\), \(3^{20 - 10}\), \(3^{10}\) (i.e., the first option \(\boldsymbol{\frac{3^{20}}{3^{10}}}\), the third option \(\boldsymbol{3^{20 - 10}}\), and the sixth option \(\boldsymbol{3^{10}}\))