Question

Use the properties of exponents and radicals to generate an equivalent expression for (256√10)551213
.(1 point)
simplify the expression

Answer 1

To simplify the expression \( (256\sqrt{10})^{551213} \), we can start by breaking it down using the properties of exponents.

1. Rewrite \( 256 \) as a power of \( 2 \):
\[
256 = 2^8
\]

2. Rewrite \( \sqrt{10} \) as a power:
\[
\sqrt{10} = 10^{1/2}
\]

3. Substitute these values into the original expression:
\[
(256\sqrt{10})^{551213} = (2^8 \cdot 10^{1/2})^{551213}
\]

4. Use the property of exponents \( (a \cdot b)^n = a^n \cdot b^n \):
\[
= (2^8)^{551213} \cdot (10^{1/2})^{551213}
\]

5. Apply the power of a power property \( (a^m)^n = a^{mn} \):
\[
= 2^{8 \cdot 551213} \cdot 10^{(1/2) \cdot 551213}
\]

6. Perform the multiplication:
\[
8 \cdot 551213 = 4409704
\]
\[
(1/2) \cdot 551213 = 275606.5
\]

7. Substitute back into the expression:
\[
= 2^{4409704} \cdot 10^{275606.5}
\]

Thus, the simplified expression is:
\[
2^{4409704} \cdot 10^{275606.5}
\]

This is the equivalent expression in its simplified form.