Power Play (Exponents) | A

Question 3

Tremendous in Ten!

Find a partner to play this game with. In 10 seconds, the person who writes a number or an expression, using only the digits 0-9 and arithmetic operations, that gives a number that is the larger between the two wins the round.

Below are some conditions that you may consider for different rounds.

(i) Exponents are not allowed. Only addition is allowed.

(ii) Exponents are not allowed. Only addition and multiplication are allowed.

(iii) Exponents are allowed. Only addition is allowed.

(iv) Exponents are allowed. Any arithmetic operation is allowed.

You can create your own conditions and/or involve more people to play together.

Solution

Understand the Question

To win a round of Tremendous in Ten!, players create the largest possible number or mathematical expression within 10 seconds under given operational constraints.
To compare large numbers or expressions:
- Express them in standard exponential form or compare total number of digits.
- When adding identical powers, factor out the common term (e.g. $10^x + 10^x = 2 \times 10^x$ ) rather than adding the exponents.
- For powers of $10$ , a larger exponent results in an exponentially larger magnitude that easily dominates smaller constant multipliers.

Step 1 · Compare $10,000,000,000,000$ and $999,999 \times 999,999$

Diagram 1

First expression: $10,000,000,000,000 = 10^{13} \quad (14 \text{ digits})$

Second expression: $999,999 \times 999,999 = (10^6 - 1)^2$

Using algebraic identity $(a - b)^2 = a^2 - 2ab + b^2$ with $a = 10^6$ and $b = 1$

\begin{aligned} (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 \quad (12 \text{ digits}) \end{aligned}

Since a $14$ -digit number is greater than a $12$ -digit number: $10,000,000,000,000 > 999,999 \times 999,999$

Step 2 · Compare $10^{1000} + 10^{1000} + 10^{1000} + 10^{1000}$ and $10^{1000000} \times 9000$

Diagram 2

Simplify the first expression: $10^{1000} + 10^{1000} + 10^{1000} + 10^{1000} = 4 \times 10^{1000}$

Simplify the second expression: $10^{1000000} \times 9000 = 9000 \times 10^{1000000}$

Comparing exponents of $10$ :

\begin{aligned} 9000 \times 10^{1000000} &= 9000 \times (10^{1000} \times 10^{999000}) \\ &= (9000 \times 10^{999000}) \times 10^{1000} \end{aligned}

Comparing coefficients of $10^{1000}$ : $9000 \times 10^{999000} > 4$

Therefore: $10^{1000000} \times 9000 > 10^{1000} + 10^{1000} + 10^{1000} + 10^{1000}$

Answer

First Comparison: $10,000,000,000,000$ is greater.

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

Common Mistakes

Adding Exponents During Addition: Mistakenly computing $10^{1000} + 10^{1000} + 10^{1000} + 10^{1000} = 10^{4000}$ . Remember that $a^m + a^m = 2a^m$ , whereas exponent addition only applies to multiplication ( $a^m \times a^n = a^{m+n}$ ).
Ignoring Base Magnitudes: Overestimating large multipliers (like $999,999 \times 999,999$ ) without checking that their total number of digits is still fewer than $10^{13}$ .

Question 3

Tremendous in Ten!

More questions in A

Tremendous in Ten!