Question

1. express the repeating decimal $0.overline{3}$ as a fraction.

a. $dfrac{1}{3}$
b. $dfrac{1}{9}$
c. $dfrac{3}{100}$
d. $dfrac{3}{10}$

1. classify each number as rational or irrational.

$9.6overline{8}$ $2.010010001...$
$sqrt{64}$ $dfrac{51}{5}$ $sqrt{6}$

rational	irrational

1. how would you classify the number 125?

a. perfect square
b. perfect cube
c. both a perfect square and a perfect cube
d. neither a perfect square nor a perfect cube

1. $f(x) = 3x^2 - 5$ find $f(2)$

$f(x)=7x + 12$ find $x$ if $f(x)=33$

1. what is the side length, $s$, of the square?

image of square with $a = 81\\ m^2$

1. solve the equation $x^2 = 10$.

a. $x = \pm\sqrt{10}$
b. $x = \sqrt{10}$
c. $x = \pm 5$
d. $x = 5$

Explanation:

Step1: Set variable for decimal

Let $x = 0.\overline{3} = 0.3333...$

Step2: Multiply by 10 to shift decimal

$10x = 3.3333...$

Step3: Subtract original equation

$10x - x = 3.333... - 0.333...$
$9x = 3$

Step4: Solve for x

$x = \frac{3}{9} = \frac{1}{3}$

Step1: Define rational/irrational rules

Rational: Terminating/repeating decimals, fractions, perfect roots. Irrational: Non-repeating/non-terminating decimals, non-perfect roots.

Step2: Classify each number

• $9.6\overline{8}$: Repeating decimal → Rational
• $2.010010001...$: Non-repeating/non-terminating → Irrational
• $\sqrt{64}=8$: Perfect root → Rational
• $-\frac{51}{5}$: Fraction → Rational
• $\sqrt{6}$: Non-perfect root → Irrational

Step1: Check perfect square

$\sqrt{125} \approx 11.18$, not integer → Not perfect square.

Step2: Check perfect cube

$\sqrt[3]{125} = 5$, integer → Perfect cube.

Step1: Substitute $x=2$ into $f(x)$

$f(2) = 3(2)^2 - 5$

Step2: Calculate exponent first

$3(4) - 5 = 12 - 5$

Step3: Compute final value

$12 - 5 = 7$

Answer:

A. $\frac{1}{3}$

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