algebra: concepts and connections
square and cube roots
simplify. show your work or no credit!
1) $\sqrt{507p^7}$
2) $\sqrt{1100x^3}$
3) $\sqrt{144x^2}$
4) $\sqrt{588n^3}$
5) $\sqrt3{32n^{13}}$
6) $\sqrt3{576b^8}$
7) $\sqrt3{500x^2}$
8) $\sqrt3{128a^8}$

Question

Accepted Answer

### Problem 1: Simplify $\boldsymbol{\sqrt{507p^7}}$
# Explanation:
## Step 1: Factor the radicand
Factor $507$ and $p^7$ into perfect squares and remaining factors.
$507 = 3 \times 13^2$ (since $13^2 = 169$ and $3 \times 169 = 507$)
$p^7 = p^{6 + 1} = (p^3)^2 \times p$ (using $a^{m+n}=a^m \times a^n$ and $(a^m)^n=a^{mn}$)

So, $\sqrt{507p^7} = \sqrt{13^2 \times 3 \times (p^3)^2 \times p}$

## Step 2: Apply square - root property
Use the property $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ ($a\geq0,b\geq0$) and $\sqrt{a^2}=|a|$. For $p^3$, if we assume $p\geq0$ (since we are dealing with real - valued square roots of expressions with even exponents in a basic algebra context), $|p^3| = p^3$.
$\sqrt{13^2 \times 3 \times (p^3)^2 \times p}=\sqrt{13^2}\times\sqrt{(p^3)^2}\times\sqrt{3p}$
$= 13\times p^3\times\sqrt{3p}$
$= 13p^3\sqrt{3p}$

# Answer:
$13p^3\sqrt{3p}$

### Problem 2: Simplify $\boldsymbol{\sqrt{1100x^3}}$
# Explanation:
## Step 1: Factor the radicand
Factor $1100$ and $x^3$ into perfect squares and remaining factors.
$1100 = 10^2\times11$ (since $10^2 = 100$ and $100\times11 = 1100$)
$x^3=x^{2 + 1}=(x)^2\times x$

So, $\sqrt{1100x^3}=\sqrt{10^2\times11\times x^2\times x}$

## Step 2: Apply square - root property
Using $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ and $\sqrt{a^2}=|a|$. Assuming $x\geq0$, $|x| = x$.
$\sqrt{10^2\times11\times x^2\times x}=\sqrt{10^2}\times\sqrt{x^2}\times\sqrt{11x}$
$= 10\times x\times\sqrt{11x}$
$= 10x\sqrt{11x}$

# Answer:
$10x\sqrt{11x}$

### Problem 3: Simplify $\boldsymbol{\sqrt{144x^2}}$
# Explanation:
## Step 1: Factor the radicand
$144 = 12^2$ and $x^2=x^2$

## Step 2: Apply square - root property
Using $\sqrt{a^2}=|a|$. Assuming $x$ is a real number, and if we consider the principal square root (in the context of basic algebra for non - negative values), $\sqrt{144x^2}=\sqrt{12^2\times x^2}=\sqrt{(12x)^2}$
If $x\geq0$, $\sqrt{(12x)^2}=12x$; if $x\lt0$, $\sqrt{(12x)^2}=- 12x$. But in the context of simplifying algebraic expressions (assuming $x$ is such that the expression is defined and we take the non - negative root for the square root of a square), we can write $\sqrt{144x^2} = 12|x|$. However, if we assume $x\geq0$ (a common assumption in basic algebra problems involving square roots of expressions with even exponents), the answer is $12x$.

# Answer:
$12x$ (assuming $x\geq0$)

### Problem 3 (Wait, the original problem has problem 3 as $\sqrt{144x^2}$, we already solved it. Let's move to problem 4: Simplify $\boldsymbol{\sqrt{588n^3}}$
# Explanation:
## Step 1: Factor the radicand
Factor $588$ and $n^3$ into perfect squares and remaining factors.
$588=4\times147 = 4\times3\times49=2^2\times3\times7^2$
$n^3=n^{2 + 1}=(n)^2\times n$

So, $\sqrt{588n^3}=\sqrt{2^2\times7^2\times3\times n^2\times n}$

## Step 2: Apply square - root property
Using $\sqrt{ab}=\sqrt{a}\times\sqrt{b}$ and $\sqrt{a^2}=|a|$. Assuming $n\geq0$, $|n| = n$.
$\sqrt{2^2\times7^2\times3\times n^2\times n}=\sqrt{2^2}\times\sqrt{7^2}\times\sqrt{n^2}\times\sqrt{3n}$
$= 2\times7\times n\times\sqrt{3n}$
$= 14n\sqrt{3n}$

# Answer:
$14n\sqrt{3n}$ (assuming $n\geq0$)

### Problem 5: Simplify $\boldsymbol{\sqrt[3]{32n^{13}}}$
# Explanation:
## Step 1: Factor the radicand
Factor $32$ and $n^{13}$ into perfect cubes and remaining factors.
$32 = 2^3\times4$ (since $2^3=8$ and $8\times4 = 32$)
$n^{13}=n^{12 + 1}=(n^4)^3\times n$ (using $a^{m + n}=a^m\times a^n$ and $(a^m)^n=a^{mn}$)

So, $\sqrt[3]{32n^{13}}=\sqrt[3]{2^3\times4\times(n^4)^3\times n}$

## Step 2: Apply cube - root property
Use the property $\sqrt[3]{ab}=\sqrt[3]{a}\times\sqrt[3]{b}$ and $\sqrt[3]{a^3}=a$.
$\sqrt[3]{2^3\times4\times(n^4)^3\times n}=\sqrt[3]{2^3}\times\sqrt[3]{(n^4)^3}\times\sqrt[3]{4n}$
$= 2\times n^4\times\sqrt[3]{4n}$
$= 2n^4\sqrt[3]{4n}$

# Answer:
$2n^4\sqrt[3]{4n}$

### Problem 6: Simplify $\boldsymbol{\sqrt[3]{576b^8}}$
# Explanation:
## Step 1: Factor the radicand
Factor $576$ and $b^8$ into perfect cubes and remaining factors.
$576=64\times9 = 4^3\times9$ (since $4^3 = 64$ and $64\times9=576$)
$b^8=b^{6+2}=(b^2)^3\times b^2$ (using $a^{m + n}=a^m\times a^n$ and $(a^m)^n=a^{mn}$)

So, $\sqrt[3]{576b^8}=\sqrt[3]{4^3\times9\times(b^2)^3\times b^2}$

## Step 2: Apply cube - root property
Use the property $\sqrt[3]{ab}=\sqrt[3]{a}\times\sqrt[3]{b}$ and $\sqrt[3]{a^3}=a$.
$\sqrt[3]{4^3\times9\times(b^2)^3\times b^2}=\sqrt[3]{4^3}\times\sqrt[3]{(b^2)^3}\times\sqrt[3]{9b^2}$
$= 4\times b^2\times\sqrt[3]{9b^2}$
$= 4b^2\sqrt[3]{9b^2}$

# Answer:
$4b^2\sqrt[3]{9b^2}$

### Problem 7: Simplify $\boldsymbol{\sqrt[3]{500x^2}}$
# Explanation:
## Step 1: Factor the radicand
Factor $500$ into perfect cubes and remaining factors.
$500 = 125\times4=5^3\times4$

So, $\sqrt[3]{500x^2}=\sqrt[3]{5^3\times4\times x^2}$

## Step 2: Apply cube - root property
Use the property $\sqrt[3]{ab}=\sqrt[3]{a}\times\sqrt[3]{b}$ and $\sqrt[3]{a^3}=a$.
$\sqrt[3]{5^3\times4\times x^2}=\sqrt[3]{5^3}\times\sqrt[3]{4x^2}$
$= 5\sqrt[3]{4x^2}$

# Answer:
$5\sqrt[3]{4x^2}$

### Problem 8: Simplify $\boldsymbol{\sqrt[3]{128a^8}}$
# Explanation:
## Step 1: Factor the radicand
Factor $128$ and $a^8$ into perfect cubes and remaining factors.
$128 = 64\times2=4^3\times2$
$a^8=a^{6 + 2}=(a^2)^3\times a^2$ (using $a^{m + n}=a^m\times a^n$ and $(a^m)^n=a^{mn}$)

So, $\sqrt[3]{128a^8}=\sqrt[3]{4^3\times2\times(a^2)^3\times a^2}$

## Step 2: Apply cube - root property
Use the property $\sqrt[3]{ab}=\sqrt[3]{a}\times\sqrt[3]{b}$ and $\sqrt[3]{a^3}=a$.
$\sqrt[3]{4^3\times2\times(a^2)^3\times a^2}=\sqrt[3]{4^3}\times\sqrt[3]{(a^2)^3}\times\sqrt[3]{2a^2}$
$= 4\times a^2\times\sqrt[3]{2a^2}$
$= 4a^2\sqrt[3]{2a^2}$

# Answer:
$4a^2\sqrt[3]{2a^2}$

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QUESTION IMAGE

Question

Explanation:

Problem 1: Simplify $\boldsymbol{\sqrt{507p^7}}$

Step 1: Factor the radicand

Step 2: Apply square - root property

Step 1: Factor the radicand

Step 2: Apply square - root property

Step 1: Factor the radicand

Step 2: Apply square - root property

Answer:

Problem 2: Simplify $\boldsymbol{\sqrt{1100x^3}}$