Question

practice test
if possible, find each of the following.
a. the principal square root of 144
b. $\sqrt{256}$
c. $-\sqrt{49}$
d. $\sqrt{400}$
e. $\sqrt{\frac{1}{100}}$
f. $\sqrt{0.000001}$
a. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. the principal square root of 144 is \\(\square\\). (simplify your answer.)
\\(\bigcirc\\) b. it is not possible to find the principal square root of 144.
b. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. $\sqrt{256} = \square$ (simplify your answer.)
\\(\bigcirc\\) b. it is not possible to find $\sqrt{256}$.
c. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. $-\sqrt{49} = \square$ (simplify your answer.)
\\(\bigcirc\\) b. it is not possible to find $-\sqrt{49}$.
d. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. $\sqrt{400} = \square$ (simplify your answer.)
\\(\bigcirc\\) b. it is not possible to find $\sqrt{400}$.
e. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. $\sqrt{\frac{1}{100}} = \square$ (simplify your answer.)
\\(\bigcirc\\) b. it is not possible to find $\sqrt{\frac{1}{100}}$.
f. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. $\sqrt{0.000001} = \square$ (simplify your answer.)
\\(\bigcirc\\) b. it is not possible to find $\sqrt{0.000001}$.

Explanation:

Part a:

Step1: Recall principal square root

The principal square root of a non - negative number \(x\) is the non - negative square root of \(x\), i.e., if \(y = \sqrt{x}\), then \(y\geq0\) and \(y^{2}=x\). For \(x = 144\), we know that \(12\times12=144\) and \(12\geq0\).

Step2: Determine the answer

So the principal square root of 144 is 12. So we choose option A and fill in 12.

Part b:

Step1: Recall square root definition

For \(\sqrt{256}\), we need to find a non - negative number \(y\) such that \(y^{2}=256\). We know that \(16\times16 = 256\) and \(16\geq0\).

Step2: Determine the answer

So \(\sqrt{256}=16\). We choose option A and fill in 16.

Part c:

Step1: Recall square root and negation

First, find \(\sqrt{49}\). We know that \(7\times7 = 49\), so \(\sqrt{49}=7\). Then, \(-\sqrt{49}\) is the negative of the square root of 49.

Step2: Determine the answer

So \(-\sqrt{49}=- 7\). We choose option A and fill in - 7.

Part d:

Step1: Recall square root definition

For \(\sqrt{400}\), we need to find a non - negative number \(y\) such that \(y^{2}=400\). We know that \(20\times20=400\) and \(20\geq0\).

Step2: Determine the answer

So \(\sqrt{400}=20\). We choose option A and fill in 20.

Part e:

Step1: Recall square root of a fraction

For \(\sqrt{\frac{1}{100}}\), we can use the property \(\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\) (where \(a\geq0,b > 0\)). Here \(a = 1\) and \(b = 100\). We know that \(\sqrt{1}=1\) and \(\sqrt{100}=10\). So \(\sqrt{\frac{1}{100}}=\frac{\sqrt{1}}{\sqrt{100}}=\frac{1}{10}=0.1\).

Step2: Determine the answer

We choose option A and fill in \(\frac{1}{10}\) (or 0.1).

Part f:

Step1: Recall square root of a decimal

We know that \(0.001\times0.001=0.000001\) because \((1\times10^{- 3})\times(1\times10^{-3})=1\times10^{-6}=0.000001\). So \(\sqrt{0.000001}=0.001\).

Step2: Determine the answer

We choose option A and fill in 0.001.

Answer:

a. A. The principal square root of 144 is \(\boldsymbol{12}\).
b. A. \(\sqrt{256}=\boldsymbol{16}\)
c. A. \(-\sqrt{49}=\boldsymbol{-7}\)
d. A. \(\sqrt{400}=\boldsymbol{20}\)
e. A. \(\sqrt{\frac{1}{100}}=\boldsymbol{\frac{1}{10}}\) (or \(0.1\))
f. A. \(\sqrt{0.000001}=\boldsymbol{0.001}\)