Simplify b³ × 3b² × 2b⁻² : Complete Exponent Multiplication

Simplify the following problem:

$b^3\times3b^2\times2b^{-2}=$

Note that there is multiplication between all terms in the expression. Thus we'll first apply the distributive property of multiplication to understand that we can separately handle the coefficients of the terms raised to powers, and the terms themselves separately. For clarity, let's handle it in steps:

$b^3\cdot3b^2\cdot2b^{-2}=3\cdot2\cdot b^3\cdot b^2\cdot b^{-2}=6\cdot b^3\cdot b^2\cdot b^{-2}$

Given that there is multiplication between all terms, we could do this. It should be noted that we can (and it's preferable to) skip the middle step, meaning:

Write directly:$b^3\cdot3b^2\cdot2b^{-2}=6\cdot b^3\cdot b^2\cdot b^{-2}$

From here on we will no longer write the multiplication sign and remember that it is conventional to simply place the terms next to each other\ place the term next to its coefficient to indicate multiplication between them,

Proceed to apply the law of exponents for multiplication of terms with identical bases:

$a^m\cdot a^n=a^{m+n}$

Note that this law applies to any number of terms being multiplied and not just two, for example for three terms with identical bases we obtain the following:

$a^m\cdot a^n\cdot a^k=a^{m+n}\cdot a^k=a^{m+n+k}$

When we used the above law of exponents twice, we can also perform the same calculation for four terms in multiplication, five and so on..

Let's return to the problem and apply the above law of exponents:

$6b^3b^2b^{-2}=6b^{3+2-2}=6b^3$

Therefore, the correct answer is A.

Important note:

Here we need to emphasize that we should always ask the question - what does the exponent apply to?

For example, in this problem the exponent applies only to the $b$ bases and not to the numbers, more clearly, in the following expression: $5c^7$The exponent applies only to $c$ and not to the number 5,

whereas when we write: $(5c)^7$The exponent applies to each of the terms in the multiplication within the parentheses,

meaning:$(5c)^7=5^7c^7$

This is actually the application of the law of exponents:

$(x\cdot y)^n=x^n\cdot y^n$

Which follows both from the meaning of parentheses and from the definition of exponents.