The expression (3a^2 b^3 c^2)^2(2a^2 b^5 c^5)^3 equals na^rb^sc^t

Question

The expression (3a^2 b^3 c^2)^2(2a^2 b^5 c^5)^3 equals na^rb^sc^t
where n, the leading coefficient, is:

and r, the exponent of a, is:

Answer 1

Hi Treena,

Let us break down the expression into two parts. We will use two properties of exponents

(a*b)^m = (a^m*b^m)

and (a^m)ⁿ = (a^mn)

Using this the first one - (3a²b³c²)²  .

Which results in (3a²)²*(b³)²*(c²)² or (3²*a^2*2)*(b^3*2)*(c^2*2) = (9*a⁴)*(b⁶)*(c⁴)

Therefore the first part of the expression becomes (9a⁴b⁶c⁴).

Now let us tackle the second part, which is (2a²b⁵c⁵)³ As before, this is equivalent to 

2³ * a^2*3 * b^5*3 *c^5*3 Or, (8 * a⁶ * b¹⁵ * c¹⁵) = (8a⁶b¹⁵c¹⁵)

Putting the two parts together, we have (9a⁴b⁶c⁴) * (8a⁶b¹⁵c¹⁵)

Now we will use another property of exponents, that is, (a^m)*(aⁿ) = a^{(m + n)}. Using this, our expression becomes,

(9 * 8) * (a⁴ * a⁶) * (b⁶ * b¹⁵) * (c⁴ * c¹⁵) = 72 * (a¹⁰) * (b²¹) * (c¹⁹)

Or, 72a¹⁰b²¹c¹⁹ = na^rb^sc^t. Comparing coefficients and exponents of like terms

Therefore, we get n = 72, r = 10, s = 21 and t = 19.

I hope this helps...