Using a suitable identity, evaluate (102)³.
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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=a3+b3+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: 1003 is one million, 23 is eight, and 3×100×2×102 simplifies to 600×102, 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.