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

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Correct answers:
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Answer 1

To simplify the expression $(256\sqrt{10}{)}^{551213}$, we can use the properties of exponents.

First, we can express $\sqrt{10}$ as ${10}^{1/2}$. Then we rewrite the original expression using the properties of exponents:

$$(256\sqrt{10}{)}^{551213}=(256\cdot {10}^{1/2}{)}^{551213}$$

Using the property $(a\cdot b{)}^m=a^m\cdot b^m$:

$$={256}^{551213}\cdot ({10}^{1/2}{)}^{551213}$$

Now, applying the exponent to the ${10}^{1/2}$:

$$={256}^{551213}\cdot {10}^{(1/2)\cdot 551213}$$

To simplify further, let's evaluate ${256}^{551213}$. Note that $256=2^8$:

$${256}^{551213}=(2^8{)}^{551213}=2^{8\cdot 551213}=2^{4409704}$$

Now, we simplify the ${10}^{(1/2)\cdot 551213}$:

$${10}^{(1/2)\cdot 551213}={10}^{275606.5}$$

Thus, the entire expression is:

$$(256\sqrt{10}{)}^{551213}=2^{4409704}\cdot {10}^{275606.5}$$

And remember that $10=2\cdot 5$, so

$${10}^{275606.5}=(2\cdot 5{)}^{275606.5}=2^{275606.5}\cdot 5^{275606.5}$$

Combining all the terms gives us:

$$=2^{4409704}\cdot 2^{275606.5}\cdot 5^{275606.5}=2^{4409704+275606.5}\cdot 5^{275606.5}$$

Calculating the exponent for $2$:

$$4409704+275606.5=4685310.5$$

So, compiling the entire expression:

$$=2^{4685310.5}\cdot 5^{275606.5}$$

This is the simplified form of the expression $(256\sqrt{10}{)}^{551213}$. If needed, we can express it back in its original format or leave it in this exponential form.