Chapter ISG, Problem 7.1.4P

a.

To determine

To find true about p and q

Answer to Problem 7.1.4P

  $p+q=p\times q$

Explanation of Solution

Given:

  $(m^p)(m^q)={(m^p)}^q$

Calculation:

According to exponential identity, a^maⁿ  =  a^m + n  for  a ≠ 0 or  m ,  n ≥ 0

  $m^{(p+q)}={(m^p)}^q$

Solving,

  $m^{(p+q)}=m^p{}^q$

As bases are same, compare power

  $p+q=p\times q$

The value of $p,q$ is 0

For example,

  $(m^0)(m^0)={(m^0)}^0$

Solving,

  $m^{(0+0)}=m^0$

So,

1=1

b.

To determine

Answer to Problem 7.1.4P

Yes, for $p,q=1$

Explanation of Solution

Given:

  $(m^p)(m^q)={(m^p)}^q$

Calculation:

The desired statement, $(m^p)(m^q)>{(m^p)}^q$ can be true if $p,q=1$

Left hand side,

  = $(m^1)(m^1)$

Solving,

  = $m^{(1+1)}$

So,

  = m² Right hand side,

  = ${(m^p)}^q$

Substituting values,

  = ${(m^1)}^1$

Solving,

  = $m^{1\times 1}$

So,

  = m²

So, left hand side> right hand side. The statement is verified.