To simplify the expression \((9 \times 10^{16})^2\), we can apply the following rules of exponents:
- When you raise a product to a power, you raise each factor to that power.
- \((a^m)^n = a^{m \cdot n}\).
Applying these rules:
\[ (9 \times 10^{16})^2 = 9^2 \times (10^{16})^2 \]
Calculating each part:
\[ 9^2 = 81 \]
For the powers of \(10\):
\[ (10^{16})^2 = 10^{16 \times 2} = 10^{32} \]
Putting it all together:
\[ (9 \times 10^{16})^2 = 81 \times 10^{32} \]
Next, we want to convert \(81 \times 10^{32}\) into proper scientific notation. In scientific notation, the coefficient should be between 1 and 10.
We can rewrite \(81\) as \(8.1 \times 10^1\). Thus:
\[ 81 \times 10^{32} = 8.1 \times 10^1 \times 10^{32} \]
Now, we can combine the powers of \(10\):
\[ = 8.1 \times 10^{1 + 32} = 8.1 \times 10^{33} \]
Final answer in Scientific Notation: \(8.1 \times 10^{33}\).
So the correct response from the options given is:
Multiply exponents to get 8.1×10^{33}
Multiply exponents to get 8 point 1 times 10 to the 33rd power.