Simplifying $\frac{64a^{12}b^6}{1}\cdot\frac{a^6}{-8 b^3}$

Question

So I was doing problems related to simplifying rational expressions and during one question I came across something I was not sure about. I had this problem after simplifying a bit:

$$\frac {64a^{12}b^6}1\cdot\frac {a^6}{-8b^3}$$

I tried cancelling the exponents of a and b, but I'm not sure whether the result is going to be this:

$$64a^6b^3$$

Or this:

$$64a^2b^2$$

Please send help

Answer 1

$$=\frac{64a^{12}b^6}{1}\cdot \frac{a^6}{-8b^3}$$ $$=\frac{64}{-8}\cdot \frac{a^{12}b^6}{1}\cdot \frac{a^6}{b^3}$$ $$=-8\cdot \frac{a^{12}a^6}{1}\cdot \frac{b^6}{b^3}$$ $$=-8\cdot \frac{a^{12+6}}{1}\cdot \frac{b^{6-3}}{1}$$ $$=-8\cdot \frac{a^{18}}{1}\cdot \frac{b^3}{1}$$ $$=-8\cdot a^{18}\cdot b^3$$ $$=-8a^{18}b^3$$

Answer 2

There are two rules when dealing with algebraic manipulation of exponents needed for this problem:

$$a^m\times a^n=a^{m+n}$$ $$\frac 1{a^n}=a^{-n}$$

For the exponents of $b$, we have that

$$b^6\times\frac 1{b^3}=b^6\times b^{-3}=b^{6-3}=b^3$$

And with $a$, we have that

$$a^{12}\times a^6=a^{12+6}=a^{18}$$

Thus, the original expression becomes

$$64a^{12}b^6\times\frac {a^6}{-8b^3}=-8a^{18}b^3$$

Answer 3

There are several ways to simplify the term $\frac{64a^{12}b^{6}}{1}\cdot\frac{a^{6}}{-8b^{3}}$

One way is to recall that when you multiply fractions, the result is the product of the numerators divided by the product of the denominators (top times top over bottom times bottom).

$\frac{64a^{12}b^{6}\cdot {a^6}}{-8b^3}$

Note that I didn't rewrite the $1$ on the bottom because $1$ times any term is that term.

Next, we can rearrange the top to combine like terms. We use the commutative and associative properties of multiplication, which say that we will receive the same result regardless of the order in which we multiply.

$64a^{12}b^{6}\cdot {a^6} = 64 \cdot a^{12}\cdot a^6\cdot b^6$

The multiplication rule of exponents tells us that $a^m \cdot a^n = a ^ {m + n}$ so $a^{12} \cdot a^6 = a ^ {12 + 6} = a ^ {18}$. The top of our equation is now simplified to $64a^{18}b^6$. The bottom of our equation is already simplified, since there are no like terms.

We now have $\frac{64a^{18}b^6}{-8b^3}$, which can be broken up into $\frac{64}{-8}\cdot a^{18}\cdot\frac{b^6}{b^3}$.

$\frac{64}{-8} = -8$

The division rule of exponents tells us that $\frac{b^m}{b^n} = b^{m-n}$ so $\frac{b^6}{b^3} = b^{6-3} = b^3$

Putting the parts back together we have $-8a^{18}b^3$.