Chapter 7.2, Problem 2P

a.

To determine

To Write: The simplified form of the expression $5x^4\cdot x^9\cdot 3x$ .

Answer to Problem 2P

The required answer is $15x^{14}$ .

Explanation of Solution

Given information:

The expression $5x^4\cdot x^9\cdot 3x$

Calculation:

  $\begin{array}{l}5x^4\cdot x^9\cdot 3x\\ =\left(5\cdot 3\right)\left(x^4\cdot x^9\cdot x\right)\end{array}$

(Commutative and associative properties of multiplication)

  $=15\left(x^4\cdot x^9\cdot x^1\right)$

(Multiply the coefficients. Write x as $x^1$ )

  $=15\left(x^{4+9+1}\right)$

(Add exponents of powers with the same base)

  $=15x^{14}$

(Simplify)

b.

To determine

To Write: The simplified form of the expression $-4c^3\cdot 7d^2\cdot 2c^{-2}$

Answer to Problem 2P

The required answer is $-56c{d}^2$ .

Explanation of Solution

Given information:

The expression $-4c^3\cdot 7d^2\cdot 2c^{-2}$

Calculation:

  $\begin{array}{l}-4c^3\cdot 7d^2\cdot 2c^{-2}\\ =\left(-4\cdot 7\cdot 2\right)\left(c^3\cdot c^{-2}\right)\left(d^2\right)\end{array}$

(Commutative and associative properties of multiplication)

  $=-56\left(c^{3+\left(-2\right)}\right)\left(d^2\right)$

(Multiply the coefficients. Add the exponents of the powers with the same base)

  $=-56c{d}^2$

(Simplify)

c.

To determine

To Write: The simplified form of the expression $j^2\cdot k^{-2}\cdot 12j$ .

Answer to Problem 2P

The required answer is $12j^3{k}^{-2}$ .

Explanation of Solution

Given information:

  $j^2\cdot k^{-2}\cdot 12j$

Calculation:

  $\begin{array}{l}{j}^2\cdot k^{-2}\cdot 12j\\ =12\left(j^2\cdot j^1\right)\left(k^{-2}\right)\end{array}$

(Commutative and associative properties of multiplication, Write $j$ as $j^1$ )

  $=12\left(j^{2+1}\right)\left(k^{-2}\right)$

(Add the exponents of the powers with the same base)

  $=12j^3{k}^{-2}$

(Simplify)

d.

To determine

To explain: How to simplify the expression $x^a\cdot x^b\cdot x^c$ .

Answer to Problem 2P

The required answer is $x^{a+b+c}$ .

Explanation of Solution

Given information:

  $x^a\cdot x^b\cdot x^c$

Calculation:

  $x^a\cdot x^b\cdot x^c$

Since all the exponents of the powers has the same base,

Therefore, add the exponents of the powers..

i.e.

  $=x^{a+b+c}$

So, the simplified expression is $x^{a+b+c}$ .