# How do you simplify (p^5r^2)^4(-7p^3r^4)^2(6pr^3)?

Dec 1, 2017

First of all, let's multiply in the outside exponents, then put the similar terms together:

#### Explanation:

We have: (p^5r^2)^4(−7p^3r^4)^2(6pr^3).

So, ${\left({p}^{5} {r}^{2}\right)}^{4} = \left({p}^{20} {r}^{8}\right)$,

(−7p^3r^4)^2 = (-7^2p^6r^8),

and $\left(6 p {r}^{3}\right)$.

Let's put the expression back together:

$\left({p}^{20} {r}^{8}\right) {\left(- 7\right)}^{2} \left({p}^{6} {r}^{8}\right) \left(6 p {r}^{3}\right)$.

Since everything is being multiplied, we can separate the factors and the constants and collect similar ones together:

p^20r^8(-7)^2(p^6r^8 (6) pr^3 = .

$\left({p}^{20} {p}^{6} p\right) \left({r}^{8} {r}^{8} {r}^{3}\right) {\left(- 7\right)}^{2} \left(6\right)$.

When you multiply numbers raised to an exponent and the bases are the same, you add the exponents . So:

$\left({p}^{20} {p}^{6} p\right) = {p}^{27}$ ,

$\left({r}^{8} {r}^{8} {r}^{3}\right) = {r}^{19}$

and ${\left(- 7\right)}^{2} \left(6\right) = 49 \cdot 6 = 294$.

Looks like we have $294 {p}^{27} {r}^{19}$.

Go through all these steps yourself, writing them down as you go, and check my arithmetic, okay?
Connie

Dec 1, 2017

The problem simplifies to
$294$ ${p}^{27}$ ${r}^{19}$

#### Explanation:

This is one big long multiplication problem.

(p^5 r^2)^4 ⋅ (−7 ${p}^{3}$ r^4)^2 ⋅($6$ $p$ ${r}^{3}$)

Step One:
Clear the parentheses by raising all the powers inside each parentheses to the power outside it.

Remember that 1 is the exponent for $- 7$ and for $6$
(not written because it is understood.) But you can always just pencil in the 1s yourself to keep things clear.

(p^5 r^2)^4 ⋅ (−7^1 ${p}^{3}$ r^4)^2 ⋅ (${6}^{1}$ $p$ r^3)^1

Step One is to raise the powers inside each parentheses to the power outside it.

To raise a power to a power, you multiply.

After you have raised each power inside each parentheses to the power outside it by multiplying them, you will have this

(p^20 r^8)⋅ (−7^2 ${p}^{6}$ r^8)  ⋅ ($6$ $p$ ${r}^{3}$)
....................

Step Two
Simplify by multiplying the amounts with like bases together

It's much easier to do this if you first collect all the factors with like bases.
Collect the numbers, the $p$'s and the $r$'s together,

($- {7}^{2}$ ⋅ 6) ×   (${p}^{20}$${p}^{6}$${p}^{1}$)× (${r}^{8}$${r}^{8}$${r}^{3}$)

To multiply exponents with like bases, you add

(49⋅6)$\left({p}^{27}\right)$$\left({r}^{19}\right)$

$294$ ${p}^{27}$ ${r}^{19}$