Question 3

We will simplify each expression and then compare their values to find out which one is greater.

Step 1 — Comparing the first pair of numbers

Let us look at the first expression. It is $10,000,000,000,000$. This number has 1 followed by 13 zeros. So, we can write it as $10^{13}$.

Now, let us look at the second expression. It is $999,999 \times 999,999$. We can write $999,999$ as $(1,000,000 - 1)$. This is $(10^6 - 1)$. So, the expression becomes $(10^6 - 1) \times (10^6 - 1)$. This is the same as $(10^6 - 1)^2$. We use the algebraic identity $(a-b)^2 = a^2 - 2ab + b^2$. Here, $a$ is $10^6$ and $b$ is $1$.

$(10^6 - 1)^2 = (10^6)^2 - 2 \times 10^6 \times 1 + 1^2$

$= 10^{12} - 2 \times 10^6 + 1$

$= 1,000,000,000,000 - 2,000,000 + 1$

$= 999,998,000,000 + 1$

$= 999,998,000,001$

Now, we compare $10,000,000,000,000$ and $999,998,000,001$. The first number has 14 digits. The second number has 12 digits. A number with more digits is always larger. So, $10,000,000,000,000$ is greater.

$\boxed{10,000,000,000,000 \text{ is greater}}$

Step 2 — Comparing the second pair of numbers

Let us look at the first expression. It is $10^{1000} + 10^{1000} + 10^{1000} + 10^{1000}$. This is the sum of four identical terms. We can write this as $4 \times 10^{1000}$.

Now, let us look at the second expression. It is $10^{1000000} \times 9000$. We can write this as $9000 \times 10^{1000000}$.

We need to compare $4 \times 10^{1000}$ and $9000 \times 10^{1000000}$. Let us compare the exponents of 10 in both expressions. For the first expression, the exponent is $1000$. For the second expression, the exponent is $1,000,000$. We know that $1,000,000$ is much larger than $1000$. This means $10^{1000000}$ is vastly larger than $10^{1000}$. Specifically, $10^{1000000} = 10^{1000} \times 10^{999000}$. So, the second expression is $9000 \times (10^{1000} \times 10^{999000})$. This is $9000 \times 10^{999000} \times 10^{1000}$. The first expression is $4 \times 10^{1000}$. Comparing the coefficients, we have $4$ versus $9000 \times 10^{999000}$. The term $9000 \times 10^{999000}$ is an extremely large number. So, $9000 \times 10^{1000000}$ is much greater.

$\boxed{10^{1000000} \times 9000 \text{ is greater}}$

Answer

First Comparison: $10,000,000,000,000$ is greater. Second Comparison: $10^{1000000} \times 9000$ is greater.