Final answer:
To simplify the inequality 100 √ 24698 ≤ 400 √ 7570246, each side is converted to scientific notation, and then their square roots are simplified, indicating that the right side of the inequality is larger, thus satisfying the inequality.
Step-by-step explanation:
The student is asking to simplify the inequality 100 √ 24698 ≤ 400 √ 7570246 to its simplest form. To do this, we should express each side in scientific notation and then simplify the square roots. First, let's convert 24698 to scientific notation, which is approximately 2.4698 x 10³. The square root of this number would then be roughly √2.4698 x 10±.5, which simplifies further to approximately 1.572 x 10±.75. Similarly, we'll convert 7570246 to scientific notation, which is approximately 7.570246 x 10⁶. Taking its square root gives us about √7.570246 x 10³, which simplifies to approximately 2.752 x 10².
After this conversion, we can see that the inequality can be rewritten as 100 x 1.572 x 10±.75 ≤ 400 x 2.752 x 10². From here, we can compare the magnitudes of each side directly, without actual calculation, understanding that on the left side, we have a smaller exponent compared to the right side. Thus, without precise calculation, it is evident that the inequality holds because the exponent on the right side is significantly larger, indicating that the quantity on the right side is much larger than the one on the left side, satisfying the inequality.