simplify each expression.
a. $\sqrt{196}$\
b. $-\sqrt{64}$\
c. $\sqrt{1}$\
d. $\sqrt{-64}$\
e. $\sqrt{\frac{81}{144}}$\
f. $\sqrt3{1728}$\
g. $\sqrt3{-343}$\
h. $\sqrt4{160000}$\
i. $\sqrt4{-16}$\
j. $(125)^{\frac{1}{3}}$\
k. $(-8)^{\frac{1}{3}}$\
l. $(81)^{\frac{1}{4}}$\
m. $(-625)^{\frac{1}{4}}$\
n. a sculpture in the shape of a cube has a volume of 1728 cubic feet. what is the side length of the block?

Question

Accepted Answer

### Part a: Simplify $\boldsymbol{\sqrt{196}}$
# Explanation:
## Step1: Recall square of 14
We know that $14	imes14 = 196$, so $\sqrt{196}$ is the number whose square is 196.
$\sqrt{196}=\sqrt{14^2}$
## Step2: Apply square - root property
For a non - negative real number $a$, $\sqrt{a^2}=a$. Since $14\geq0$, we have $\sqrt{14^2} = 14$.
# Answer:
$14$

### Part b: Simplify $\boldsymbol{-\sqrt{64}}$
# Explanation:
## Step1: Recall square of 8
We know that $8	imes8=64$, so $\sqrt{64}=\sqrt{8^2}$.
## Step2: Apply square - root property and consider the negative sign
For a non - negative real number $a$, $\sqrt{a^2}=a$. So $\sqrt{8^2} = 8$, and then $-\sqrt{64}=-8$.
# Answer:
$-8$

### Part c: Simplify $\boldsymbol{\sqrt{1}}$
# Explanation:
## Step1: Recall square of 1
We know that $1	imes1 = 1$, so $\sqrt{1}=\sqrt{1^2}$.
## Step2: Apply square - root property
For a non - negative real number $a$, $\sqrt{a^2}=a$. Since $1\geq0$, $\sqrt{1^2}=1$.
# Answer:
$1$

### Part d: Simplify $\boldsymbol{\sqrt{- 64}}$
# Explanation:
The square root of a negative number is not a real number. In the set of real numbers, for the square root function $y = \sqrt{x}$, the domain is $x\geq0$. Since $-64<0$, $\sqrt{-64}$ is not a real number (it is an imaginary number, and in the complex number system, $\sqrt{-64}=8i$, but if we are working in the real number system, it is undefined).
# Answer:
Not a real number (or in complex numbers: $8i$)

### Part e: Simplify $\boldsymbol{\sqrt{\frac{81}{144}}}$
# Explanation:
## Step1: Use the property of square roots
We know that for non - negative real numbers $a$ and $b$ ($b
eq0$), $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$. So $\sqrt{\frac{81}{144}}=\frac{\sqrt{81}}{\sqrt{144}}$.
## Step2: Simplify the numerator and the denominator
We know that $\sqrt{81}=\sqrt{9^2}=9$ and $\sqrt{144}=\sqrt{12^2}=12$. So $\frac{\sqrt{81}}{\sqrt{144}}=\frac{9}{12}$.
## Step3: Simplify the fraction
We can simplify $\frac{9}{12}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 3. $\frac{9\div3}{12\div3}=\frac{3}{4}$.
# Answer:
$\frac{3}{4}$

### Part f: Simplify $\boldsymbol{\sqrt[3]{1728}}$
# Explanation:
## Step1: Recall cube of 12
We know that $12	imes12	imes12=12^3 = 1728$.
## Step2: Apply cube - root property
For any real number $a$, $\sqrt[3]{a^3}=a$. So $\sqrt[3]{12^3}=12$.
# Answer:
$12$

### Part g: Simplify $\boldsymbol{\sqrt[3]{-343}}$
# Explanation:
## Step1: Recall cube of - 7
We know that $(-7)	imes(-7)	imes(-7)=(-7)^3=-343$.
## Step2: Apply cube - root property
For any real number $a$, $\sqrt[3]{a^3}=a$. So $\sqrt[3]{(-7)^3}=-7$.
# Answer:
$-7$

### Part h: Simplify $\boldsymbol{\sqrt[4]{160000}}$
# Explanation:
## Step1: Factor 160000
We can write $160000 = 16	imes10000=2^4	imes10^4=(2	imes10)^4 = 20^4$.
## Step2: Apply fourth - root property
For a non - negative real number $a$, $\sqrt[4]{a^4}=a$. So $\sqrt[4]{20^4}=20$.
# Answer:
$20$

### Part i: Simplify $\boldsymbol{\sqrt[4]{-16}}$
# Explanation:
In the set of real numbers, the fourth root of a negative number is not defined. The domain of the function $y = \sqrt[4]{x}$ is $x\geq0$ (because we are dealing with real - valued functions). Since $-16<0$, $\sqrt[4]{-16}$ is not a real number (in the complex number system, $\sqrt[4]{-16}=\sqrt{2}(1 + i),\sqrt{2}(-1 + i),\sqrt{2}(-1 - i),\sqrt{2}(1 - i)$).
# Answer:
Not a real number (or in complex numbers: $\pm\sqrt{2}(1\pm i)$)

### Part j: Simplify $\boldsymbol{(125)^{\frac{1}{3}}}$
# Explanation:
## Step1: Recall the definition of rational exponents
We know that $a^{\frac{1}{n}}=\sqrt[n]{a}$. So $(125)^{\frac{1}{3}}=\sqrt[3]{125}$.
## Step2: Recall cube of 5
We know that $5	imes5	imes5 = 5^3=125$. So $\sqrt[3]{125}=\sqrt[3]{5^3}$.
## Step3: Apply cube - root property
For any real number $a$, $\sqrt[3]{a^3}=a$. So $\sqrt[3]{5^3}=5$.
# Answer:
$5$

### Part k: Simplify $\boldsymbol{(-8)^{\frac{1}{3}}}$
# Explanation:
## Step1: Recall the definition of rational exponents
We know that $a^{\frac{1}{n}}=\sqrt[n]{a}$. So $(-8)^{\frac{1}{3}}=\sqrt[3]{-8}$.
## Step2: Recall cube of - 2
We know that $(-2)	imes(-2)	imes(-2)=(-2)^3=-8$.
## Step3: Apply cube - root property
For any real number $a$, $\sqrt[3]{a^3}=a$. So $\sqrt[3]{(-2)^3}=-2$.
# Answer:
$-2$

### Part l: Simplify $\boldsymbol{(81)^{\frac{1}{4}}}$
# Explanation:
## Step1: Recall the definition of rational exponents
We know that $a^{\frac{1}{n}}=\sqrt[n]{a}$. So $(81)^{\frac{1}{4}}=\sqrt[4]{81}$.
## Step2: Recall fourth power of 3
We know that $3	imes3	imes3	imes3 = 3^4=81$.
## Step3: Apply fourth - root property
For a non - negative real number $a$, $\sqrt[4]{a^4}=a$. So $\sqrt[4]{3^4}=3$.
# Answer:
$3$

### Part m: Simplify $\boldsymbol{(-625)^{\frac{1}{4}}}$
# Explanation:
In the set of real numbers, the fourth root of a negative number is not defined. The domain of the function $y = \sqrt[4]{x}$ is $x\geq0$ (for real - valued functions). Since $-625<0$, $(-625)^{\frac{1}{4}}$ is not a real number (in the complex number system, $(-625)^{\frac{1}{4}} = 5i, - 5i,5, - 5$ (but considering the principal root in complex numbers, it can be represented in different forms)).
# Answer:
Not a real number (or in complex numbers: $\pm5,\pm5i$)

### Part n: Side length of the cube
# Explanation:
## Step1: Recall the volume formula of a cube
The volume $V$ of a cube with side length $s$ is given by the formula $V=s^3$.
## Step2: Substitute the given volume into the formula
We are given that $V = 1728$ cubic feet. So we have the equation $s^3=1728$.
## Step3: Solve for $s$
We need to find the cube root of 1728. We know that $12^3=12	imes12	imes12 = 1728$. So $s=\sqrt[3]{1728}=\sqrt[3]{12^3}$.
## Step4: Apply cube - root property
For any real number $a$, $\sqrt[3]{a^3}=a$. So $\sqrt[3]{12^3}=12$.
# Answer:
The side length of the cube is 12 feet.

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QUESTION IMAGE

Question

Explanation:

Part a: Simplify $\boldsymbol{\sqrt{196}}$

Step1: Recall square of 14

Step2: Apply square - root property

Step1: Recall square of 8

Step2: Apply square - root property and consider the negative sign

Step1: Recall square of 1

Step2: Apply square - root property

Answer:

Part b: Simplify $\boldsymbol{-\sqrt{64}}$