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:

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:

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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 - (3a^{2}b^{3}c^{2})^{2 } .

Which results in (3a^{2})^{2}*(b^{3})^{2}*(c^{2})^{2 }or (3^{2}*a^{2*2})*(b^{3*2})*(c^{2*2}) = (9*a^{4})*(b^{6})*(c^{4})

Therefore the first part of the expression becomes (9a^{4}b^{6}c^{4}).

Now let us tackle the second part, which is (2a^{2}b^{5}c^{5})^{3} As before, this is equivalent to

2^{3} * a^{2*3} * b^{5*3} *c^{5*3} Or, (8 * a^{6} * b^{15} * c^{15}) = (8a^{6}b^{15}c^{15})

Putting the two parts together, we have (9a^{4}b^{6}c^{4}) * (8a^{6}b^{15}c^{15})

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

(9 * 8) * (a^{4} * a^{6}) * (b^{6} * b^{15}) * (c^{4} * c^{15}) = 72 * (a^{10}) * (b^{21}) * (c^{19})

Or, 72a^{10}b^{21}c^{19} = na^{r}b^{s}c^{t}. Comparing coefficients and exponents of like terms

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

I hope this helps...

Nikunj P.

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