Find the value of 70³ + 20³ − 90³.

The correct answer is – 378000.

We need to find the value of 70³ + 20³ − 90³.

First, notice a key relationship among the three bases:
70 + 20 − 90 = 0

This is the crucial observation! We can use the algebraic identity for when the sum of three numbers equals zero.

The Identity:
If a + b + c = 0, then:

a^3+b^3+c^3=3abc

Why does this identity work?
We know the general factorisation:

a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)

When a + b + c = 0, the right-hand side becomes 0, so:

a^3+b^3+c^3-3abc=0

\Rightarrow a^3+b^3+c^3=3abc

Applying the Identity:
Let a = 70, b = 20, c = −90.
Check: 70 + 20 + (−90) = 0 ✓

So:

{70}^3+{20}^3+(-90){}^3=3\times 70\times 20\times (-90)

Which is the same as:

{70}^3+{20}^3-{90}^3=3\times 70\times 20\times (-90)

Calculating Step by Step:

Step 1: 3\times 70=210

Step 2: 210\times 20=4200

Step 3: 4200\times (-90)=-378000

Therefore:

{70}^3+{20}^3-{90}^3=-378000