Chapter 1.2, Problem 3E

(a)

To determine

To fill: The blank in the statement “Using exponential notation, we can write $\sqrt[3]{5}$ as ____”.

Answer to Problem 3E

The complete statement is “Using exponential notation, we can write $\sqrt[3]{5}$ as $\underset{\_}{5^{\frac{1}{3}}}$”.

Explanation of Solution

Formula used:

Formula for nth root:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

Where, a is any real number called as base and n is an integer.

Calculation:

The given expression is $\sqrt[3]{5}$.

Use the above formula and write the exponential notation as shown below.

$\sqrt[3]{5}=5^{\frac{1}{3}}$

Thus, the complete statement is “Using exponential notation, we can write $\sqrt[3]{5}$ as $\underset{\_}{5^{\frac{1}{3}}}$”.

(b)

To determine

To fill: The blank in the statement “Using radicals, we can write $5^{\frac{1}{2}}$ as _____”.

Answer to Problem 3E

The complete statement is “Using radicals, we can write $5^{\frac{1}{2}}$ as $\underset{\_}{\sqrt{5}}$”.

Explanation of Solution

The given expression is $5^{\frac{1}{2}}$.

Use the formula mentioned in sub part (a) and write the given exponential expression in the radical form as shown below.

$5^{\frac{1}{2}}=\sqrt[2]{5}=\sqrt{5}$

Thus, the complete statement is “Using radicals, we can write $5^{\frac{1}{2}}$ as $\underset{\_}{\sqrt{5}}$”.

(c)

To determine

Whether there is any difference between the values of $\sqrt{5^2}$ and ${\left(\sqrt{5}\right)}^2$.

Explanation of Solution

Formula used:

Formula for nth root:

$\sqrt[n]{a}=a^{\frac{1}{n}}$

Where, a is any real number called as base and n is an integer.

Calculation:

Consider the expression, $\sqrt{5^2}$.

Using the above formula and simplify the expression as shown below.

$\begin{array}{c}\sqrt{5^2}=\sqrt[2]{5^2}\\ =5^{\frac{2}{2}}\\ =5\end{array}$

Thus, the value of $\sqrt{5^2}$ is 5.

Now, consider the expression ${\left(\sqrt{5}\right)}^2$.

Use the above formula and simplify the expression as follows.

$\begin{array}{c}{\left(\sqrt{5^{}}\right)}^2={\left(\sqrt[2]{5}\right)}^2\\ ={\left(5^{\frac{1}{2}}\right)}^2\\ =5^{\frac{1}{2}\cdot 2}\\ =5\end{array}$

Therefore, it can be clearly observed that there is no difference between the values of $\sqrt{5^2}$ and ${\left(\sqrt{5}\right)}^2$.