practice: simplifying radicals with variables cals. show all work! 2. $\sqrt{m^3}$ 4. $\sqrt{25x^6}$ 6. $\sqrt{300y^4}$ 8. $\sqrt{128c^5}$ 10. $\sqrt{81u^2v}$ 12. $\sqrt{100x^3y}$ 14. $\sqrt{96c^4d^2}$ 16. $\sqrt{56m^2n^4p^3}$ 18. $\sqrt{45x^2y^5z^8}$ 20. $\sqrt{98x^4y^6z^2}$ $\copyright$ gina wilson (all things algebra$^\text{\textregistered}$, llc), 2012 - 2017

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Accepted Answer

Let's solve each problem one by one. We'll use the property of square roots: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ (where $a \geq 0$ and $b \geq 0$) and $\sqrt{x^n} = x^{\frac{n}{2}}$ (for $x \geq 0$).

### Problem 2: $\boldsymbol{\sqrt{m^3}}$
## Step 1: Factor the exponent
Rewrite $m^3$ as $m^2 \cdot m$. So, $\sqrt{m^3} = \sqrt{m^2 \cdot m}$.
## Step 2: Apply the square root property
Using $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, we get $\sqrt{m^2} \cdot \sqrt{m}$. Since $\sqrt{m^2} = m$ (for $m \geq 0$), this simplifies to $m\sqrt{m}$.
# Answer: $m\sqrt{m}$

### Problem 4: $\boldsymbol{\sqrt{25x^6}}$
## Step 1: Apply the square root property to each factor
We know that $\sqrt{25} = 5$ and $\sqrt{x^6} = x^{\frac{6}{2}} = x^3$ (using $\sqrt{x^n} = x^{\frac{n}{2}}$).
## Step 2: Multiply the results
So, $\sqrt{25x^6} = \sqrt{25} \cdot \sqrt{x^6} = 5 \cdot x^3 = 5x^3$.
# Answer: $5x^3$

### Problem 6: $\boldsymbol{\sqrt{300y^4}}$
## Step 1: Factor 300
Factor 300 as $100 \cdot 3$. So, $\sqrt{300y^4} = \sqrt{100 \cdot 3 \cdot y^4}$.
## Step 2: Apply the square root property
Using $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, we get $\sqrt{100} \cdot \sqrt{3} \cdot \sqrt{y^4}$. We know that $\sqrt{100} = 10$ and $\sqrt{y^4} = y^{\frac{4}{2}} = y^2$.
## Step 3: Multiply the results
So, this simplifies to $10 \cdot y^2 \cdot \sqrt{3} = 10y^2\sqrt{3}$.
# Answer: $10y^2\sqrt{3}$

### Problem 8: $\boldsymbol{\sqrt{128c^5}}$
## Step 1: Factor 128 and $c^5$
Factor 128 as $64 \cdot 2$ and $c^5$ as $c^4 \cdot c$. So, $\sqrt{128c^5} = \sqrt{64 \cdot 2 \cdot c^4 \cdot c}$.
## Step 2: Apply the square root property
Using $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, we get $\sqrt{64} \cdot \sqrt{2} \cdot \sqrt{c^4} \cdot \sqrt{c}$. We know that $\sqrt{64} = 8$ and $\sqrt{c^4} = c^{\frac{4}{2}} = c^2$.
## Step 3: Multiply the results
So, this simplifies to $8 \cdot c^2 \cdot \sqrt{2} \cdot \sqrt{c} = 8c^2\sqrt{2c}$.
# Answer: $8c^2\sqrt{2c}$

### Problem 10: $\boldsymbol{\sqrt{81u^2v}}$
## Step 1: Apply the square root property to each factor
We know that $\sqrt{81} = 9$, $\sqrt{u^2} = u$ (for $u \geq 0$), and $\sqrt{v}$ remains as is.
## Step 2: Multiply the results
So, $\sqrt{81u^2v} = \sqrt{81} \cdot \sqrt{u^2} \cdot \sqrt{v} = 9 \cdot u \cdot \sqrt{v} = 9u\sqrt{v}$.
# Answer: $9u\sqrt{v}$

### Problem 12: $\boldsymbol{\sqrt{100x^3y}}$
## Step 1: Factor $x^3$
Rewrite $x^3$ as $x^2 \cdot x$. So, $\sqrt{100x^3y} = \sqrt{100 \cdot x^2 \cdot x \cdot y}$.
## Step 2: Apply the square root property
Using $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, we get $\sqrt{100} \cdot \sqrt{x^2} \cdot \sqrt{x} \cdot \sqrt{y}$. We know that $\sqrt{100} = 10$ and $\sqrt{x^2} = x$ (for $x \geq 0$).
## Step 3: Multiply the results
So, this simplifies to $10 \cdot x \cdot \sqrt{x} \cdot \sqrt{y} = 10x\sqrt{xy}$.
# Answer: $10x\sqrt{xy}$

### Problem 14: $\boldsymbol{\sqrt{96c^4d^2}}$
## Step 1: Factor 96
Factor 96 as $16 \cdot 6$. So, $\sqrt{96c^4d^2} = \sqrt{16 \cdot 6 \cdot c^4 \cdot d^2}$.
## Step 2: Apply the square root property
Using $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, we get $\sqrt{16} \cdot \sqrt{6} \cdot \sqrt{c^4} \cdot \sqrt{d^2}$. We know that $\sqrt{16} = 4$, $\sqrt{c^4} = c^{\frac{4}{2}} = c^2$, and $\sqrt{d^2} = d$ (for $d \geq 0$).
## Step 3: Multiply the results
So, this simplifies to $4 \cdot c^2 \cdot d \cdot \sqrt{6} = 4c^2d\sqrt{6}$.
# Answer: $4c^2d\sqrt{6}$

### Problem 16: $\boldsymbol{\sqrt{56m^2n^4p^3}}$
## Step 1: Factor 56 and $p^3$
Factor 56 as $4 \cdot 14$ and $p^3$ as $p^2 \cdot p$. So, $\sqrt{56m^2n^4p^3} = \sqrt{4 \cdot 14 \cdot m^2 \cdot n^4 \cdot p^2 \cdot p}$.
## Step 2: Apply the square root property
Using $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, we get $\sqrt{4} \cdot \sqrt{14} \cdot \sqrt{m^2} \cdot \sqrt{n^4} \cdot \sqrt{p^2} \cdot \sqrt{p}$. We know that $\sqrt{4} = 2$, $\sqrt{m^2} = m$ (for $m \geq 0$), $\sqrt{n^4} = n^{\frac{4}{2}} = n^2$, and $\sqrt{p^2} = p$ (for $p \geq 0$).
## Step 3: Multiply the results
So, this simplifies to $2 \cdot m \cdot n^2 \cdot p \cdot \sqrt{14} \cdot \sqrt{p} = 2mn^2p\sqrt{14p}$.
# Answer: $2mn^2p\sqrt{14p}$

### Problem 18: $\boldsymbol{\sqrt{45x^2y^5z^8}}$
## Step 1: Factor 45 and $y^5$
Factor 45 as $9 \cdot 5$ and $y^5$ as $y^4 \cdot y$. So, $\sqrt{45x^2y^5z^8} = \sqrt{9 \cdot 5 \cdot x^2 \cdot y^4 \cdot y \cdot z^8}$.
## Step 2: Apply the square root property
Using $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, we get $\sqrt{9} \cdot \sqrt{5} \cdot \sqrt{x^2} \cdot \sqrt{y^4} \cdot \sqrt{y} \cdot \sqrt{z^8}$. We know that $\sqrt{9} = 3$, $\sqrt{x^2} = x$ (for $x \geq 0$), $\sqrt{y^4} = y^{\frac{4}{2}} = y^2$, and $\sqrt{z^8} = z^{\frac{8}{2}} = z^4$.
## Step 3: Multiply the results
So, this simplifies to $3 \cdot x \cdot y^2 \cdot z^4 \cdot \sqrt{5} \cdot \sqrt{y} = 3xy^2z^4\sqrt{5y}$.
# Answer: $3xy^2z^4\sqrt{5y}$

### Problem 20: $\boldsymbol{\sqrt{98x^4y^6z^2}}$
## Step 1: Factor 98
Factor 98 as $49 \cdot 2$. So, $\sqrt{98x^4y^6z^2} = \sqrt{49 \cdot 2 \cdot x^4 \cdot y^6 \cdot z^2}$.
## Step 2: Apply the square root property
Using $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, we get $\sqrt{49} \cdot \sqrt{2} \cdot \sqrt{x^4} \cdot \sqrt{y^6} \cdot \sqrt{z^2}$. We know that $\sqrt{49} = 7$, $\sqrt{x^4} = x^{\frac{4}{2}} = x^2$, $\sqrt{y^6} = y^{\frac{6}{2}} = y^3$, and $\sqrt{z^2} = z$ (for $z \geq 0$).
## Step 3: Multiply the results
So, this simplifies to $7 \cdot x^2 \cdot y^3 \cdot z \cdot \sqrt{2} = 7x^2y^3z\sqrt{2}$.
# Answer: $7x^2y^3z\sqrt{2}$

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QUESTION IMAGE

Question

Explanation:

Problem 2: $\boldsymbol{\sqrt{m^3}}$

Step 1: Factor the exponent

Step 2: Apply the square root property

Answer:

Problem 4: $\boldsymbol{\sqrt{25x^6}}$

Step 1: Apply the square root property to each factor

Step 2: Multiply the results