Q. 20 - Intermediate Algebra

Question

In the following exercises, estimate each root between two consecutive whole numbers. 

ⓐ $\sqrt{55}$ ⓑ $\sqrt[3]{119}$ 

Step-by-Step Solution

Answer

Part (a)$\Rightarrow \sqrt{55}=7<\sqrt{55}<8$

Part (b)$\Rightarrow \sqrt[3]{119}=4<\sqrt[3]{119}<5$

Part (a) Step 1. Given information

(a)$\sqrt{55}$

Estimate each root between two consecutive whole numbers.

Step 2. Explanation

Think of the perfect square numbers closest to $55$.

 Make a small table of these perfect squares and their squares roots.

$\begin{array}{cc}\text{Number}& \text{Square root}\\ \begin{array}{l}49\\ 64\end{array}& \begin{array}{l}7\\ 8\end{array}\end{array}$

Locate $55$ between two consecutive perfect squares.

$49<55<64$

$\sqrt{55}$ is between their square roots.

$7<\sqrt{55}<8$

Part (b) Step 3. Given information

(b)$\sqrt[3]{119}$

Estimate each root between two consecutive whole numbers.

Step 4. Explanation

Similarly, we locate $119$between two perfect cube numbers.

$\begin{array}{cc}\text{Number}& \text{Cube root}\\ \begin{array}{l}64\\ 125\end{array}& \begin{array}{l}4\\ 5\end{array}\end{array}$

Locate $119$between two consecutive perfect cubes.

$\Rightarrow 64<71<125$

$\sqrt[3]{119}$ is between their cube roots.

$4<\sqrt[3]{119}<5$