Questions: Use exponent rules to write the following expression in the form 3^n and simplify. Do not evaluate. 9 * (1/9)^(-2) =

Solution Steps

To simplify the expression \(9 \cdot \left(\frac{1}{9}\right)^{-2}\) using exponent rules, we first express 9 as a power of 3, which is \(3^2\). The term \(\left(\frac{1}{9}\right)^{-2}\) can be rewritten as \(9^2\) because \(\left(\frac{1}{9}\right)^{-2} = 9^2\). Then, express \(9^2\) as \((3^2)^2\), which simplifies to \(3^4\). Finally, combine the exponents of the base 3: \(3^2 \cdot 3^4 = 3^{2+4} = 3^6\).

The number 9 can be expressed as a power of 3:
\[ 9 = 3^2 \]

The expression \(\left(\frac{1}{9}\right)^{-2}\) can be rewritten as:
\[ \left(\frac{1}{9}\right)^{-2} = 9^2 \]
Since \(9 = 3^2\), we have:
\[ 9^2 = (3^2)^2 = 3^4 \]

Now, combine the exponents of the base 3 in the expression \(9 \cdot \left(\frac{1}{9}\right)^{-2}\):
\[ 3^2 \cdot 3^4 = 3^{2+4} = 3^6 \]

Final Answer