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

Question

.(1 point)

Answer 1

To simplify the expression $(256\sqrt{10}{)}^{551213}$, we can first break it down:

1. Recognize that $256$ can be expressed as a power of 2: $$256=2^8$$ So we can rewrite the expression as: $$(256\sqrt{10}{)}^{551213}=(2^8\sqrt{10}{)}^{551213}$$

2. The square root can be rewritten as a fractional exponent: $$\sqrt{10}={10}^{1/2}$$ Therefore, we have: $$(2^8\sqrt{10}{)}^{551213}=(2^8\cdot {10}^{1/2}{)}^{551213}$$

3. Now we apply the exponent to both components in the parentheses: $$=(2^8{)}^{551213}\cdot ({10}^{1/2}{)}^{551213}$$

4. Using the property of exponents $(a^m{)}^n=a^{m\cdot n}$: $$=2^{8\cdot 551213}\cdot {10}^{(1/2)\cdot 551213}$$

5. Now we can calculate the exponents: $$8\cdot 551213=4409704$$ and $$\frac{1}{2}\cdot 551213=275606.5$$

Thus, the equivalent expression is: $$2^{4409704}\cdot {10}^{275606.5}$$

This final expression, $2^{4409704}\cdot {10}^{275606.5}$, is the simplified equivalent of the original expression $(256\sqrt{10}{)}^{551213}$.