in exercises 35-42, simplify the expression. (see example 5.)35. $sqrt4{81y^8}$36. $sqrt3{8w^9}$37. $sqrt3{64r^3t^6}$38. $sqrt4{16a^8b^{12}}$39. $\frac{sqrt5{m^{10}}}{m^3}$40. $sqrt6{\frac{k^{18}}{16z^4}}$▶41. $sqrt{\frac{g^6h}{h^7}}$42. $sqrt8{\frac{n^2p^{-1}}{n^{18}p^7}}$in exercises 43-48, perform the indicated operation and write the answer in simplest form. assume all variables are positive. (see example 6.)▶43. $12sqrt3{y}+9sqrt3{y}$44. $11sqrt{2z}-5sqrt{2z}$45. $3x^{7/2}-5x^{7/2}$46. $7m^{7/3}+3m^{7/3}$47. $(16w^{10})^{1/4}+2w(w^6)^{1/4}$48. $sqrt3{32p^{10}}-9p^2sqrt3{4p^4}$4 error analysis in exercises 49 and 50, describe and correct the error in simplifying the expression.49.$sqrt6{\frac{64}{w^6}}=\frac{sqrt6{64}}{sqrt6{w^6}}$$=\frac{sqrt6{2^6}}{sqrt6{w^6}}$$=\frac{2}{w}$50.$3sqrt3{12y}+5sqrt3{12y}=(3+5)sqrt3{24y}$$=8sqrt3{24y}$$=8sqrt3{8cdot3y}$$=8cdot2sqrt3{3y}$$=16sqrt3{3y}$51. open-ended write two variable expressions involving radicals, one that needs absolute value when simplifying and one that does not need absolute value. justify your answers.52. college prep when each expression is simplified, which expressions are like radicals? select all that apply.a $sqrt3{125x}$b $sqrt4{80x}$c $sqrt3{625x}$d $sqrt3{40x}$e $sqrt4{16x}$f $sqrt3{135x}$

Question

Accepted Answer

### First, solving Exercises 35-42 (simplify radicals):
# Explanation:
## Step1: Rewrite radicand as perfect 4th power
$\sqrt[4]{81y^8} = \sqrt[4]{3^4 \cdot (y^2)^4}$
## Step2: Extract perfect 4th power terms
$\sqrt[4]{3^4 \cdot (y^2)^4} = 3y^2$

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## Step1: Rewrite radicand as perfect 3rd power
$\sqrt[3]{8w^9} = \sqrt[3]{2^3 \cdot (w^3)^3}$
## Step2: Extract perfect 3rd power terms
$\sqrt[3]{2^3 \cdot (w^3)^3} = 2w^3$

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## Step1: Rewrite radicand as perfect 3rd power
$\sqrt[3]{64r^9t^6} = \sqrt[3]{4^3 \cdot (r^3)^3 \cdot (t^2)^3}$
## Step2: Extract perfect 3rd power terms
$\sqrt[3]{4^3 \cdot (r^3)^3 \cdot (t^2)^3} = 4r^3t^2$

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## Step1: Rewrite radicand as perfect 4th power
$\sqrt[4]{16a^8b^{12}} = \sqrt[4]{2^4 \cdot (a^2)^4 \cdot (b^3)^4}$
## Step2: Extract perfect 4th power terms
$\sqrt[4]{2^4 \cdot (a^2)^4 \cdot (b^3)^4} = 2a^2b^3$

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## Step1: Combine radicals into single fraction
$\frac{\sqrt[5]{m^{10}}}{\sqrt[5]{m^5}} = \sqrt[5]{\frac{m^{10}}{m^5}}$
## Step2: Simplify exponent inside radical
$\sqrt[5]{m^{10-5}} = \sqrt[5]{m^5}$
## Step3: Extract perfect 5th power term
$\sqrt[5]{m^5} = m$

---
## Step1: Split radical into fraction of radicals
$\sqrt[6]{\frac{k^{18}}{16z^4}} = \frac{\sqrt[6]{k^{18}}}{\sqrt[6]{16z^4}}$
## Step2: Simplify numerator and rewrite denominator
$\frac{k^{3}}{(2^4z^4)^{1/6}} = \frac{k^3}{2^{2/3}z^{2/3}} = \frac{k^3}{\sqrt[3]{4z^2}}$
## Step3: Rationalize the denominator
$\frac{k^3\sqrt[3]{2z}}{\sqrt[3]{4z^2} \cdot \sqrt[3]{2z}} = \frac{k^3\sqrt[3]{2z}}{2z}$

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## Step1: Split radical into fraction of radicals
$\sqrt{\frac{g^6h}{h^7}} = \frac{\sqrt{g^6h}}{\sqrt{h^7}}$
## Step2: Simplify exponents of radicals
$\frac{g^3\sqrt{h}}{h^{3}\sqrt{h}}$
## Step3: Cancel common radical terms
$\frac{g^3}{h^3}$

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## Step1: Split radical into fraction of radicals
$\sqrt[8]{\frac{n^2p^{-1}}{n^{18}p^7}} = \sqrt[8]{n^{2-18}p^{-1-7}}$
## Step2: Simplify exponents inside radical
$\sqrt[8]{n^{-16}p^{-8}}$
## Step3: Rewrite with positive exponents
$\sqrt[8]{\frac{1}{n^{16}p^8}} = \frac{1}{n^2p}$

# Answer:
35. $3y^2$
36. $2w^3$
37. $4r^3t^2$
38. $2a^2b^3$
39. $m$
40. $\frac{k^3\sqrt[3]{2z}}{2z}$
41. $\frac{g^3}{h^3}$
42. $\frac{1}{n^2p}$

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### Next, solving Exercises 43-48 (combine like radicals/terms):
# Explanation:
## Step1: Combine like radical terms
$12\sqrt[3]{y} + 9\sqrt[3]{y} = (12+9)\sqrt[3]{y}$
## Step2: Simplify the coefficient
$21\sqrt[3]{y}$

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## Step1: Combine like radical terms
$11\sqrt{2z} - 5\sqrt{2z} = (11-5)\sqrt{2z}$
## Step2: Simplify the coefficient
$6\sqrt{2z}$

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## Step1: Combine like exponential terms
$3x^{7/2} - 5x^{7/2} = (3-5)x^{7/2}$
## Step2: Simplify the coefficient
$-2x^{7/2}$

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## Step1: Combine like exponential terms
$7m^{7/3} + 3m^{7/3} = (7+3)m^{7/3}$
## Step2: Simplify the coefficient
$10m^{7/3}$

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## Step1: Simplify each exponential term
$(16w^{10})^{1/4} = 16^{1/4}w^{10/4} = 2w^{5/2}$; $2w(w^6)^{1/4} = 2w \cdot w^{6/4} = 2w^{5/2}$
## Step2: Combine like terms
$2w^{5/2} + 2w^{5/2} = 4w^{5/2}$

---
## Step1: Simplify each radical term
$\sqrt[3]{32p^{10}} = \sqrt[3]{16 \cdot 2 \cdot p^9 \cdot p} = 2p^3\sqrt[3]{4p}$; $9p^2\sqrt[3]{4p^4} = 9p^2 \cdot p\sqrt[3]{4p} = 9p^3\sqrt[3]{4p}$
## Step2: Combine like radical terms
$2p^3\sqrt[3]{4p} - 9p^3\sqrt[3]{4p} = (2-9)p^3\sqrt[3]{4p}$
## Step3: Simplify the coefficient
$-7p^3\sqrt[3]{4p}$

# Answer:
43. $21\sqrt[3]{y}$
44. $6\sqrt{2z}$
45. $-2x^{7/2}$
46. $10m^{7/3}$
47. $4w^{5/2}$
48. $-7p^3\sqrt[3]{4p}$

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### Error Analysis (Exercises 49-50):
# Brief Explanations:
#### Exercise 49:
The error is that $\sqrt[6]{64} = 2$, but since the root index (6) is even, the result of $\sqrt[6]{w^6}$ must include an absolute value to ensure non-negativity (even roots yield non-negative results, and $w$ could be negative).
#### Exercise 50:
The error is incorrectly combining the radicands when adding like radicals. Like radicals have identical radicands and root indices, so we only add the coefficients, not combine the radicands.

# Answer:
49. Correct simplification: $\frac{2}{|w|}$
50. Correct simplification:
$3\sqrt[3]{12y} + 5\sqrt[3]{12y} = (3+5)\sqrt[3]{12y} = 8\sqrt[3]{12y}$

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### Exercise 51 (Open-Ended):
# Brief Explanations:
- An expression needing absolute value: Even-indexed roots of variable terms where the simplified exponent is odd (to ensure the result is non-negative, as even radicals cannot produce negative values).
- An expression not needing absolute value: Odd-indexed roots (odd radicals preserve the sign of the input, so no absolute value is required) or even-indexed roots where the simplified variable exponent is even.

# Answer:
1. Needs absolute value: $\sqrt[4]{x^8y^5} = x^2|y|\sqrt[4]{y}$. Justification: The 4th root (even index) of $y^5$ simplifies to $|y|\sqrt[4]{y}$; absolute value ensures the result is non-negative since even roots only yield non-negative outputs.
2. No absolute value needed: $\sqrt[3]{x^6y^9} = x^2y^3$. Justification: The 3rd root (odd index) preserves the sign of the original variables, so no absolute value is required.

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### Exercise 52 (College Prep):
# Brief Explanations:
Like radicals have the same root index and identical simplified radicands. Simplify each option:
- $\boldsymbol{\sqrt[3]{125x}} = 5\sqrt[3]{x}$
- $\boldsymbol{\sqrt[4]{80x}} = 2\sqrt[4]{5x}$
- $\boldsymbol{\sqrt[3]{625x}} = 5\sqrt[3]{5x}$
- $\boldsymbol{\sqrt[3]{40x}} = 2\sqrt[3]{5x}$
- $\boldsymbol{\sqrt[4]{16x}} = 2\sqrt[4]{x}$
- $\boldsymbol{\sqrt[3]{135x}} = 3\sqrt[3]{5x}$

# Answer:
C. $\sqrt[3]{625x}$, D. $\sqrt[3]{40x}$, F. $\sqrt[3]{135x}$

QUESTION IMAGE

Question

Explanation:

First, solving Exercises 35-42 (simplify radicals):

Step1: Rewrite radicand as perfect 4th power

Step2: Extract perfect 4th power terms

Step1: Rewrite radicand as perfect 3rd power

Step2: Extract perfect 3rd power terms

Step1: Rewrite radicand as perfect 3rd power

Step2: Extract perfect 3rd power terms

Step1: Rewrite radicand as perfect 4th power

Step2: Extract perfect 4th power terms

Step1: Combine radicals into single fraction

Step2: Simplify exponent inside radical

Step3: Extract perfect 5th power term

Step1: Split radical into fraction of radicals

Step2: Simplify numerator and rewrite denominator

Step3: Rationalize the denominator

Step1: Split radical into fraction of radicals

Step2: Simplify exponents of radicals

Step3: Cancel common radical terms

Step1: Split radical into fraction of radicals

Step2: Simplify exponents inside radical

Step3: Rewrite with positive exponents

Step1: Combine like radical terms

Step2: Simplify the coefficient

Step1: Combine like radical terms

Step2: Simplify the coefficient

Step1: Combine like exponential terms

Step2: Simplify the coefficient

Step1: Combine like exponential terms

Step2: Simplify the coefficient

Step1: Simplify each exponential term

Step2: Combine like terms

Step1: Simplify each radical term

Step2: Combine like radical terms

Step3: Simplify the coefficient

Exercise 49:

Exercise 50:

Answer:

Next, solving Exercises 43-48 (combine like radicals/terms):