We will use the algebraic identity for the cube of a sum, which states that (a+b)³ = a³ + b³ + 3ab(a+b). This identity will help us expand and evaluate the expression.

We calculate each term separately: 100³ is one million, 2³ is eight, and 3 × 100 × 2 × 102 simplifies to 600 × 102, which is 61,200.

Using a suitable identity, evaluate (102)³.

Answer: $(102)^3 = 1{,}061{,}208$

Step-by-step solution

Step 1: Rewrite the expression

To use an algebraic identity, we can rewrite the number 102 as a sum of two numbers, 100 and 2. This makes it easier to apply the identity for $(a+b)^3$ .

Step 2: Apply the identity for (a+b)³

We will use the algebraic identity for the cube of a sum, which states that $(a+b)^3 = a^3 + b^3 + 3ab(a+b)$ . This identity will help us expand and evaluate the expression.

Step 3: Substitute values into the identity

Now, we substitute $a=100$ and $b=2$ into the identity. This gives us the expanded form of the expression, which we can then simplify.

Step 4: Calculate the terms

We calculate each term separately: $100^3$ is one million, $2^3$ is eight, and $3 \times 100 \times 2 \times 102$ simplifies to $600 \times 102$ , which is 61,200.

Step 5: Sum the calculated terms

Finally, we add all the calculated terms together: $1{,}000{,}000 + 8 + 61{,}200$ . This sum gives us the final value of $(102)^3$ .

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