Question

1. $\sqrt{256} = \underline{quadquad}$
2. $\sqrt{198x^3y^4} = \underline{quadquad}$
3. $-\sqrt{144} = \underline{quadquad}$

try some multiple choice.

1. $\sqrt{405x^7y^6z^3} = \underline{quadquad}$

a. $9x^3y^3z\sqrt{5xz}$
b. $3x^3y^3z\sqrt{5xz}$
c. $5x^3y^3z\sqrt{9xz}$
d. $9x^6y^6z\sqrt{5xz}$

1. $-2\sqrt{\frac{24}{484}} = \underline{quadquad}$

a. $\frac{-2\sqrt{6}}{121}$
b. $\frac{-2\sqrt{6}}{11}$
c. $\frac{-6\sqrt{2}}{11}$
d. $\frac{-4\sqrt{6}}{121}$

1. $\sqrt{432} = \underline{quadquad}$

a. $3\sqrt{12}$
b. $13\sqrt{2}$
c. $3\sqrt{102}$
d. $12\sqrt{3}$

1. simplify. answers must not include negative exponents.

a. $\frac{4a^4b^3c^{-2}}{(a^{-2}b^2c)^3}$
b. $(2x^2y^3)(5xy^2)$
c. $(2x^2y)^2(3xy^2)^3$

1. write the equation of a line that is parallel to $2x - 5y = 20$ and contains the point $(-5, 3)$.
2. what is the range of the function $y = 2x - 7$ if the domain is ${-2, 0, 4}$?
3. a) write a function rule for the arithmetic sequence listed below. $a_n = \underline{quadquad}$

${2, -4, -10, -16, \dots}$
b) what is the $25^{\text{th}}$ term of the sequence?

Explanation:

Step1: Find square root of 256

$\sqrt{256} = \sqrt{16^2} = 16$

Step2: Factor radicand for $\sqrt{198x^3y^4}$

$\sqrt{198x^3y^4} = \sqrt{9 \times 22 \times x^2 \times x \times (y^2)^2} = 3xy^2\sqrt{22x}$

Step3: Find negative square root of 144

$-\sqrt{144} = -\sqrt{12^2} = -12$

Step4: Simplify $\sqrt{405x^7y^6z^3}$

$\sqrt{405x^7y^6z^3} = \sqrt{81 \times 5 \times x^6 \times x \times (y^3)^2 \times z^2 \times z} = 9x^3y^3z\sqrt{5xz}$

Step5: Simplify $-2\sqrt{\frac{24}{484}}$

$-2\sqrt{\frac{24}{484}} = -2 \times \frac{\sqrt{4 \times 6}}{22} = -2 \times \frac{2\sqrt{6}}{22} = \frac{-2\sqrt{6}}{11}$

Step6: Simplify $\sqrt{432}$

$\sqrt{432} = \sqrt{144 \times 3} = 12\sqrt{3}$

Step7: Simplify $\frac{4a^4b^3c^{-2}}{(a^{-2}b^2c)^3}$

First expand denominator: $(a^{-2}b^2c)^3 = a^{-6}b^6c^3$
Then apply exponent rules: $\frac{4a^4b^3c^{-2}}{a^{-6}b^6c^3} = 4a^{4-(-6)}b^{3-6}c^{-2-3} = 4a^{10}b^{-3}c^{-5} = \frac{4a^{10}}{b^3c^5}$

Step8: Simplify $(2x^2y^{-3})(5xy^2)$

Apply exponent rules: $2 \times 5 \times x^{2+1}y^{-3+2} = 10x^3y^{-1} = \frac{10x^3}{y}$

Step9: Simplify $(2x^2y)^2(3xy^{-2})^3$

First expand each term:
$(2x^2y)^2 = 4x^4y^2$, $(3xy^{-2})^3 = 27x^3y^{-6}$
Multiply and simplify: $4x^4y^2 \times 27x^3y^{-6} = 108x^{4+3}y^{2-6} = 108x^7y^{-4} = \frac{108x^7}{y^4}$

Step10: Find parallel line equation

Rewrite $2x-5y=20$ in slope-intercept form: $y = \frac{2}{5}x - 4$, slope $m=\frac{2}{5}$
Use point-slope form $y-y_1=m(x-x_1)$ with $(-5,3)$:
$y-3 = \frac{2}{5}(x+5)$
Simplify to slope-intercept: $y = \frac{2}{5}x + 2 + 3 = \frac{2}{5}x + 5$
Or standard form: $2x - 5y = -25$

Step11: Calculate function range

For $y=2x-7$, domain $\{-2,0,4\}$:

• $x=-2$: $y=2(-2)-7=-4-7=-11$
• $x=0$: $y=2(0)-7=-7$
• $x=4$: $y=2(4)-7=8-7=1$

Range: $\{-11, -7, 1\}$

Step12: Find arithmetic sequence rule

Sequence $\{2,-4,-10,-16,...\}$, first term $a_1=2$, common difference $d=-4-2=-6$
Rule: $a_n = a_1 + (n-1)d = 2 + (n-1)(-6) = 2 -6n +6 = 8-6n$

Step13: Find 25th term of sequence

Substitute $n=25$ into $a_n=8-6n$:
$a_{25}=8-6(25)=8-150=-142$

Answer:

1. $16$
2. $3xy^2\sqrt{22x}$
3. $-12$
4. A. $9x^3y^3z\sqrt{5xz}$
5. B. $\frac{-2\sqrt{6}}{11}$
6. D. $12\sqrt{3}$

22.
a. $\frac{4a^{10}}{b^3c^5}$
b. $\frac{10x^3}{y}$
c. $\frac{108x^7}{y^4}$

1. $y=\frac{2}{5}x+5$ (or $2x-5y=-25$)
2. $\{-11, -7, 1\}$

25.
a. $a_n=8-6n$
b. $-142$