Question 14

We will use algebraic identities to simplify each expression and then compare their values.

Step 1 — Evaluate expressions involving squares

We use the algebraic identity $a^2 - b^2 = (a - b)(a + b)$. This identity helps us calculate the difference of two squares.

For (iii) $67^2 - 66^2$: Here, $a = 67$ and $b = 66$.

$67^2 - 66^2 = (67 - 66)(67 + 66)$

$= (1)(133)$

$\boxed{133}$

For (iv) $43^2 - 42^2$: Here, $a = 43$ and $b = 42$.

$43^2 - 42^2 = (43 - 42)(43 + 42)$

$= (1)(85)$

$\boxed{85}$

Step 2 — Evaluate expressions involving cubes

We use the algebraic identity $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$. This identity helps us calculate the difference of two cubes.

For (i) $67^3 - 66^3$: Here, $a = 67$ and $b = 66$.

$67^3 - 66^3 = (67 - 66)(67^2 + 67 \times 66 + 66^2)$

$= (1)(4489 + 4422 + 4356)$

$= 13267$

$\boxed{13267}$

For (ii) $43^3 - 42^3$: Here, $a = 43$ and $b = 42$.

$43^3 - 42^3 = (43 - 42)(43^2 + 43 \times 42 + 42^2)$

$= (1)(1849 + 1806 + 1764)$

$= 5419$

$\boxed{5419}$

Step 3 — Compare all the values

Let us list the values we found for each expression: (i) $67^3 - 66^3 = \mathbf{13267}$ (ii) $43^3 - 42^3 = \mathbf{5419}$ (iii) $67^2 - 66^2 = \mathbf{133}$ (iv) $43^2 - 42^2 = \mathbf{85}$

Comparing these numbers, we see that $\mathbf{13267}$ is the largest value. This means $67^3 - 66^3$ is the greatest expression.

Step 4 — Explain the general reasoning

Let us consider the general forms of these expressions, where $n$ is a positive integer. For the difference of consecutive squares, like $(n+1)^2 - n^2$: Using the identity $a^2 - b^2 = (a-b)(a+b)$, this simplifies to $(n+1-n)(n+1+n) = 1 \times (2n+1) = 2n+1$.

For the difference of consecutive cubes, like $(n+1)^3 - n^3$: Using the identity $a^3 - b^3 = (a-b)(a^2+ab+b^2)$, this simplifies to $(n+1-n)((n+1)^2 + (n+1)n + n^2)$. This further simplifies to $1 \times (n^2+2n+1 + n^2+n + n^2) = 3n^2 + 3n + 1$.

We can see that the difference of cubes, $3n^2 + 3n + 1$, has an $n^2$ term. The difference of squares, $2n + 1$, only has an $n$ term. For positive values of $n$, the $n^2$ term grows much faster than the $n$ term. This means that the difference of cubes will always be much larger than the difference of squares for the same $n$.

Also, both $2n+1$ and $3n^2+3n+1$ increase as $n$ increases. For the cube differences, $67^3 - 66^3$ uses $n=66$, while $43^3 - 42^3$ uses $n=42$. Since $66 > 42$, $67^3 - 66^3$ is greater. For the square differences, $67^2 - 66^2$ uses $n=66$, while $43^2 - 42^2$ uses $n=42$. Since $66 > 42$, $67^2 - 66^2$ is greater. Comparing the largest cube difference ($13267$) with the largest square difference ($133$), the cube difference is clearly much greater. Therefore, $67^3 - 66^3$ is the greatest value.

Answer

(i) $67^3 - 66^3 = 13267$ (ii) $43^3 - 42^3 = 5419$ (iii) $67^2 - 66^2 = 133$ (iv) $43^2 - 42^2 = 85$