Chapter R, Problem 1PRE

Many expressions in algebra look similar but the methods used to simplify the expressions may be different. For Exercises 1-14, simplify each expression. Assume that the variables are restricted so that each expression is defined.

1.

1. ${64}^{1/2}$
2. ${64}^{1/3}$
3. ${64}^{2/3}$
4. ${64}^{-1}$
5. $-{64}^{1/2}$
6. ${\left(-64\right)}^{1/2}$
7. ${\left(-64\right)}^{2/3}$
8. ${\left(-64\right)}^{-2/3}$

To determine

The simplified form of the expression ${\left(64\right)}^{\frac{1}{2}}$ .

Answer to Problem 1PRE

The simplified form of the expression ${\left(64\right)}^{\frac{1}{2}}$ is $8$ .

Explanation of Solution

Given information:

The expression is ${\left(64\right)}^{\frac{1}{2}}$ .

Consider the expression ${\left(64\right)}^{\frac{1}{2}}$ .

Now simplify to get,

  $\begin{array}{c}{\left(64\right)}^{\frac{1}{2}}={\left(8\cdot 8\right)}^{\frac{1}{2}}\\ ={\left(8^2\right)}^{\frac{1}{2}}\\ =8\end{array}$

Therefore, the simplified form of the expression ${\left(64\right)}^{\frac{1}{2}}$ is $8$ .

To determine

The simplified form of the expression ${\left(64\right)}^{\frac{1}{3}}$ .

Answer to Problem 1PRE

The simplified form of the expression ${\left(64\right)}^{\frac{1}{3}}$ is $4$ .

Explanation of Solution

Given information:

The expression is ${\left(64\right)}^{\frac{1}{3}}$ .

Consider the expression ${\left(64\right)}^{\frac{1}{3}}$ .

  $\begin{array}{c}{\left(64\right)}^{\frac{1}{3}}={\left(4\cdot 4\cdot 4\right)}^{\frac{1}{3}}\\ ={\left(4^3\right)}^{\frac{1}{3}}\\ =4\end{array}$

Therefore, the simplified form of the expression ${\left(64\right)}^{\frac{1}{3}}$ is $4$ .

To determine

The simplified form of the expression ${\left(64\right)}^{\frac{2}{3}}$ .

Answer to Problem 1PRE

The simplified form of the expression ${\left(64\right)}^{\frac{2}{3}}$ is $16$ .

Explanation of Solution

Given information:

The expression is ${\left(64\right)}^{\frac{2}{3}}$ .

Consider the expression ${\left(64\right)}^{\frac{2}{3}}$ .

  $\begin{array}{c}{\left(64\right)}^{\frac{2}{3}}={\left(4\cdot 4\cdot 4\right)}^{\frac{2}{3}}\\ ={\left(4^3\right)}^{\frac{2}{3}}\\ =4^2\\ =16\end{array}$

Therefore, the simplified form of the expression ${\left(64\right)}^{\frac{2}{3}}$ is $16$ .

To determine

The simplified form of the expression ${\left(64\right)}^{-1}$ .

Answer to Problem 1PRE

The simplified form of the expression ${\left(64\right)}^{-1}$ is $\frac{1}{64}$ .

Explanation of Solution

Given information:

The expression is ${\left(64\right)}^{-1}$ .

Consider the expression ${\left(64\right)}^{-1}$ .

  ${\left(64\right)}^{-1}=\frac{1}{64}$

To determine

The simplified form of the expression $-{\left(64\right)}^{\frac{1}{2}}$ .

Answer to Problem 1PRE

The simplified form of the expression $-{\left(64\right)}^{\frac{1}{2}}$ is $-8$ .

Explanation of Solution

Given information:

The expression is $-{\left(64\right)}^{\frac{1}{2}}$ .

Consider the expression $-{\left(64\right)}^{\frac{1}{2}}$ .

Now simplify to get,

  $\begin{array}{c}-{\left(64\right)}^{\frac{1}{2}}=-{\left(8\cdot 8\right)}^{\frac{1}{2}}\\ =-{\left(8^2\right)}^{\frac{1}{2}}\\ =-8\end{array}$

To determine

The simplified form of the expression ${\left(-64\right)}^{\frac{1}{2}}$ .

Answer to Problem 1PRE

The expression ${\left(-64\right)}^{\frac{1}{2}}$ is not defined

Explanation of Solution

Given information:

The expression is ${\left(-64\right)}^{\frac{1}{2}}$ .

Consider the expression ${\left(-64\right)}^{\frac{1}{2}}$ .

The value $\left(-64\right)$ is negative so its root cannot be calculated on the real number line.

To determine

The simplified form of the expression ${\left(-64\right)}^{\frac{2}{3}}$ .

Answer to Problem 1PRE

The simplified form of the expression ${\left(-64\right)}^{\frac{2}{3}}$ is $16$ .

Explanation of Solution

Given information:

The expression is ${\left(-64\right)}^{\frac{2}{3}}$ .

Consider the expression ${\left(-64\right)}^{\frac{2}{3}}$ .

  $\begin{array}{c}{\left(-64\right)}^{\frac{2}{3}}={\left(-4\cdot 4\cdot 4\right)}^{\frac{2}{3}}\\ ={\left(-4^3\right)}^{\frac{2}{3}}\\ ={\left(-1\right)}^{\frac{2}{3}}{4}^2\\ =16\end{array}$

To determine

The simplified form of the expression ${\left(-64\right)}^{\frac{-2}{3}}$ .

Answer to Problem 1PRE

The simplified form of the expression ${\left(-64\right)}^{\frac{-2}{3}}$ is $\frac{1}{16}$ .

Explanation of Solution

Given information:

The expression is ${\left(-64\right)}^{\frac{-2}{3}}$ .

Consider the expression ${\left(-64\right)}^{\frac{-2}{3}}$ .

  $\begin{array}{c}{\left(-64\right)}^{\frac{-2}{3}}=\frac{1}{{\left(-1\right)}^{\frac{2}{3}}{\left(64\right)}^{\frac{2}{3}}}\\ =\frac{1}{16}\end{array}$