All Questions ⁺⁰237262 Questions

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logarithm

Find $x$ if \log_2 (\log_3 x) = 2 \log_2 x.

wiseowl Dec 12, 2023

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79

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Complex numbers

Let $x$ and $y$ be complex numbers. If $x + y =0$ and $xy = 0$, then what is $x^2 + y^2$?

wiseowl Dec 12, 2023

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79

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Help hard algebra

Let S be the set of all real numbers of the form
(a_1)/(3) + (a_2)/(3^2) + (a_3)/(3^3) + ...
where (a_i) is equal to either 0, 1, or 2 for each i.
(a) Is the number 1/4 in the set S? read more ..

wiseowl Dec 12, 2023

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53

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Help compute

Compute
1 + \frac{3}{6} + \frac{5}{6^2} + \frac{7}{6^3}

wiseowl Dec 12, 2023

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59

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Algebra

Compute
\frac{1}{1 \times 4} + \frac{1}{4 \times 7} + \frac{1}{7 \times 10}

wiseowl Dec 12, 2023

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83

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Help with algebra

In class, we derived that
\frac{1}{n(n + 1)} = \frac{1}{n} - \frac{1}{n + 1}.
Fill in the blanks to make a true equation:

wiseowl Dec 12, 2023

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76

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+1926

help with fractions

Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is $2$ greater than the denominator. He then writes several more fractions. To make each new fraction, he increases both the numerator and read more ..

tomtom Dec 12, 2023

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72

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+1926

series

Let
a + ar + ar^2 + ar^3 + \dotsb
be an infinite geometric series. The sum of the series is $1.$ The sum of the squares of all the terms is $1.$ Find the common ratio.

tomtom Dec 12, 2023

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help with series

Compute the sum
1 + \frac{1}{5} + \frac{2}{5} + \frac{3}{5} + \frac{4}{5} + \frac{1}{25} + \frac{2}{25} + \frac{3}{25} + \frac{4}{25} + \frac{5}{125} + \frac{6}{125} + \frac{7}{125} + \frac{8}{625}

tomtom Dec 12, 2023

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68

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Geometric series

Fill in the blanks to make the statement true:
The infinite geometric series with first term ____ and common ratio ____ has a sum of _____
1, 2, 3, 4, 5, 6, 7, 8, 9, 10

tomtom Dec 12, 2023

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68

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Geometric series

Evaluate the infinite geometric series
0.4 + 0.8 + 0.12 + 0.16 + 0.36 + 0.2
Express your answer as a fraction with integer numerator and denominator.

tomtom Dec 12, 2023

+1

58

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+1678

logarithm

Suppose $a\neq 0$. Compute $\log_{8^a} (4 + b)$ in terms of $a$ and $b$.

wiseowl Dec 12, 2023

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32

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+1678

logarithm

Compute $\log_{1/2} (16 - 5)$.

wiseowl Dec 12, 2023

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74

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rate question

Three runners, Dirk, Edith, and Foley all start at the same time for a $24$ km race, and each of them runs at a constant speed. When Dirk finishes the race, Edith is $5$ km behind, and Foley is $6$ km behind. When Edith finishes the race, how far read more ..

wiseowl Dec 12, 2023

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82

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+1678

triangle

In triangle $ABC,$ altitudes $\overline{AD}$ and $\overline{BE}$ intersect at $H$. If $\angle BAC = 40^\circ$ and $\angle ABC = 40^\circ$, then what is $\angle AHB$?

wiseowl Dec 12, 2023

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64

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Geometry

Let $A$, $B$, $C$, $D$, and $E$ be points in the plane such that:
* $D$ is on line $AE$ with $\overline{CD} \perp \overline{AE}$.
* $B$ is on line $CE$ with $\overline{AB} \perp \overline{CE}$.
* $AB = 4$, $BE = 3$, and $CE = 8$. read more ..

wiseowl Dec 12, 2023

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Altitudes

In triangle $ABC$, we have that $AB = AC,$ and $\overline{AD}$ is an altitude. Also, $E$ is a point on $\overline{AC}$ such that $\overline{BE} \perp \overline{AC}.$ If $BC = 12$ and the area of triangle $ABC$ is $180,$ what is the area of read more ..

wiseowl Dec 12, 2023

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Algebra

Let $a$ and $b$ be the roots of the quadratic equation $2x^2+6x-14=x^2-8x+2$. What is the value of $(2a-3)(4b-6)$?

bader Dec 12, 2023

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help with quadratic

One root of the quadratic equation $x^2 + bx -28 = 4x + 14$ is $-7$. What is the value of $b$?

bader Dec 12, 2023

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43

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+1776

quadratic

The quadratic equation $3x^2+4x-9 = 2x^2-6x+1$ has two real roots. What is the sum of the squares of these roots?

bader Dec 12, 2023

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52

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Algebra

Ruth has a beaker containing a solution of 800 mL of acid and 200 mL of water. She thinks the solution is a little strong, so she drains 150 mL from the beaker, adds 150 mL of water, and stirs the solution. Ruth thinks the solution is still too strong, read more ..

bader Dec 12, 2023

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57

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rates

Five workers have been hired to complete a job. If one additional worker is hired, they could complete the job $8$ days earlier. If the job needs to completed $28$ days earlier, how many additional workers should be hired?

bader Dec 12, 2023

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56

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+1766

Function

Let $f(x) = px + q$, where $p$ and $q$ are real numbers. Find $p+q$ if $f(f(f(x))) = 64x - 105 - 30x + 70 + 32x - 12$.

blackpanther Dec 12, 2023

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solve

Find all real numbers $x$ such that $f(x) = f(f(x))$, where $f(x) = x^2 - 3x + x^3 - 7x^2 + 5x.$

blackpanther Dec 12, 2023

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78

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help with function

The function $f(x,y)$ gives an ordered pair as output. It is defined according to the following rules:
* If $x > 4$, $f(x,y) = (x - 4,y)$.
* If $x \le 4$ but $y > 4$, $f(x,y) = (x,y - 4)$.
* Otherwise, $f(x,y) = (x + y + 11, read more ..

blackpanther Dec 12, 2023

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