Chapter 7.3, Problem 1P

(a)

To determine

To simplify: The expression ${\left(p^5\right)}^4$ .

Answer to Problem 1P

The simplified form of the expression ${\left(p^5\right)}^4$ is $p^{20}$ .

Explanation of Solution

Given information:

The given expression is ${\left(p^5\right)}^4$ .

Calculation:

Solve the expression.

  $\begin{array}{c}{\left(p^5\right)}^4=p^{5\times 4}\\ =p^{20}\end{array}$

Conclusion:

Therefore, the simplified form of the expression ${\left(p^5\right)}^4$ is $p^{20}$ .

(b)

To determine

To simplify: The expression ${\left(p^4\right)}^5$ .

Answer to Problem 1P

The simplified form of the expression ${\left(p^4\right)}^5$ is $p^{20}$ .

Explanation of Solution

Given information:

The given expression is ${\left(p^4\right)}^5$ .

Calculation:

Solve the expression.

  $\begin{array}{c}{\left(p^4\right)}^5=p^{4\times 5}\\ =p^{20}\end{array}$

Conclusion:

Therefore, the simplified form of the expression ${\left(p^4\right)}^5$ is $p^{20}$ .

(c)

To determine

To simplify: The expression ${\left(p^{\frac{1}{2}}\right)}^{\frac{1}{4}}$ .

Answer to Problem 1P

The simplified form of the expression ${\left(p^{\frac{1}{2}}\right)}^{\frac{1}{4}}$ is $p^{\frac{1}{8}}$ .

Explanation of Solution

Given information:

The given expression is ${\left(p^{\frac{1}{2}}\right)}^{\frac{1}{4}}$ .

Calculation:

Solve the expression.

  $\begin{array}{c}{\left(p^{\frac{1}{2}}\right)}^{\frac{1}{4}}=p^{\frac{1}{2}\times \frac{1}{4}}\\ =p^{\frac{1}{8}}\end{array}$

Conclusion:

Therefore, the simplified form of the expression ${\left(p^{\frac{1}{2}}\right)}^{\frac{1}{4}}$ is $p^{\frac{1}{8}}$ .

(d)

To determine

To simplify: The expression ${\left(p^{\frac{1}{4}}\right)}^{\frac{1}{2}}$ .

Answer to Problem 1P

The simplified form of the expression ${\left(p^{\frac{1}{4}}\right)}^{\frac{1}{2}}$ is $p^{\frac{1}{8}}$ .

Explanation of Solution

Given information:

The given expression is ${\left(p^{\frac{1}{4}}\right)}^{\frac{1}{2}}$ .

Calculation:

Solve the expression.

  $\begin{array}{c}{\left(p^{\frac{1}{4}}\right)}^{\frac{1}{2}}=p^{\frac{1}{4}\times \frac{1}{2}}\\ =p^{\frac{1}{8}}\end{array}$

Conclusion:

Therefore, the simplified form of the expression ${\left(p^{\frac{1}{4}}\right)}^{\frac{1}{2}}$ is $p^{\frac{1}{8}}$ .

(e)

To determine

To check: Whether the expression ${\left(a^m\right)}^n={\left(a^n\right)}^m$ is true for all integers $m$ and $n$ or not.

Answer to Problem 1P

Yes, the expression ${\left(a^m\right)}^n={\left(a^n\right)}^m$ is true for all integers $m$ and $n$ .

Explanation of Solution

Given information:

The given expression is ${\left(a^m\right)}^n={\left(a^n\right)}^m$ . The integers are $m$ and $n$

Calculation:

Let’s consider the value of $m$ is $-1$ and $n=2$ . Consider the left hand side of the expression and substitute the value.

  $\begin{array}{c}{\left(a^{-1}\right)}^2=a^{-1\times 2}\\ =a^{-2}\end{array}$

Consider the left hand side of the expression and substitute the value.

  $\begin{array}{c}{\left(a^2\right)}^{-1}=a^{2\times \left(-1\right)}\\ =a^{-2}\end{array}$

From the above, it is clear that the left hand side of the expression is equal to the right hand side of the expression. Therefore, the expression ${\left(a^m\right)}^n={\left(a^n\right)}^m$ is true for all integers $m$ and $n$ .

Conclusion:

Therefore, the expression ${\left(a^m\right)}^n={\left(a^n\right)}^m$ is true for all integers $m$ and $n$ .