Chapter 7.3, Problem 90HP

a.

To determine

To check if the statement is always, sometimes or never true.

Answer to Problem 90HP

Sometimes

Explanation of Solution

Given:

Statement: $x^2=x^{\frac{1}{2}}$

Consider the given expression $x^2=x^{\frac{1}{2}}$ ,

The square of a term and the square root of the term is the same only when the term is 1.

Conclusion:

Therefore, the expression is only for true for $x=1.$

b.

To determine

To check if the statement is always, sometimes or never true.

Answer to Problem 90HP

Sometimes

Explanation of Solution

Given:

Statement: $x^{-2}=x^{\frac{1}{2}}$

Consider the statement $x^{-2}=x^{\frac{1}{2}},$

The square of inverse and the square root are the same only if the term is 1.

Conclusion:

Therefore, the given statement is true only for $x=1.$

c.

To determine

To check if the statement is always, sometimes or never true.

Answer to Problem 90HP

Sometimes

Explanation of Solution

Given:

Statement: $x^{\frac{1}{3}}=x^{\frac{1}{2}}$

Consider the expression $x^{\frac{1}{3}}=x^{\frac{1}{2}},$

The square root and the cube root of a term is the same only when the term is 1.

Conclusion:

Therefore, the given expression is true only when $x=1.$

d.

To determine

To check if the statement is always, sometimes or never true.

Answer to Problem 90HP

Always

Explanation of Solution

Given:

Statement: $\sqrt{x}=x^{\frac{1}{2}}$

Consider the given statement $\sqrt{x}=x^{\frac{1}{2}},$

The statement is true since it is the definition of square root.

Conclusion:

Therefore, the given statement is always true.

e.

To determine

To check if the statement is always, sometimes or never true.

Answer to Problem 90HP

Always

Explanation of Solution

Given:

Statement: ${\left(x^{\frac{1}{2}}\right)}^2=x$

Consider the given statement ${\left(x^{\frac{1}{2}}\right)}^2=x,$

The statement is always true since the exponents cancel each other and gives x raised to the power 1.

Conclusion:

Therefore, the given statement is always true.

f.

To determine

To check if the statement is always, sometimes or never true.

Answer to Problem 90HP

Sometimes

Explanation of Solution

Given:

Statement: $x^{\frac{1}{2}}\cdot x^2=x$

Consider the given statement $x^{\frac{1}{2}}\cdot x^2=x,$

The given statement is true only when $x=1$ since whenever 1 is squared or taken square root the value remains the same.

Conclusion:

Therefore, the given statement is true only for $x=1.$