Chapter ISG, Problem 8.3.4P

a.

To determine

Answer to Problem 8.3.4P

  $x^{-4p^2+4}$ is the simplified expression

Explanation of Solution

Given:

  $x^{4p+1}{(x^{1-2p})}^{2p+3}$

Calculation:

The given expression is $x^{4p+1}{(x^{1-2p})}^{2p+3}$

Using laws of exponent,

  = $x^{4p+1}{(x)}^{(2p+3)(1-2p)}$$\because$

  ${(a^m)}^n=a^{mn}$

Solving powers we get,

  = $x^{4p+1}{x}^{-4p^2+3-4p}$

Again as bases are same power adds,

  = $x^{4p+1-4p^2+3-4p}$$\because a^m.a^n=a^{m+n}$

So,

  = $x^{-4p^2+4}$

Hence, $x^{-4p^2+4}$ is the simplified expression.

b.

To determine

To find value of p

Answer to Problem 8.3.4P

p is 2, -2 for x=1

Explanation of Solution

Given:

Simplified expression $x^{-4p^2+4}$

Calculation:

For the value of x to be 1,

We know that $a^0=1$

Substituting values,

1= $x^{-4p^2+4}$

Now,

  $x^0=x^{-4p^2+4}$

As bases are equal, compare power.

  $0=-4p^2+4^{}$

Now,

  $\begin{array}{l}4p^2-4=0\\ $ {(2p)}^2-2_{}^2=0$\end{array}$

Using identity $a^2-b^2=(a+b)(a-b)$

  $(2p-2)(2p+2)=0$

Solving for p ,

  $p=2,p=-2$

Hence, value of p is 2, -2 for x=1 for all values of x