All Questions ⁺⁰237179 Questions

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Algebra

For each of the following functions, determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain. Select all the properties that apply.
f(x) = sqrt{10 - x} + \sqrt{x + 10}

coIinqiu Mar 11, 2025

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Algebra

For each of the following functions, determine if the function is increasing, decreasing, even, odd, and/or invertible on its natural domain. Select all the properties that apply.
f(x) = x^3 - 3x^2 + 10x - 15

coIinqiu Mar 11, 2025

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+95

Algebra

Find all real numbers a and b such that
a + b = 14
a^3 + b^3 = 812 + a^2 + b^2

coIinqiu Mar 11, 2025

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+95

Algebra

Factor (ab + ac + bc)^3 - a^3 b^3 - a^3 c^3 - b^3 c^3 + (a^2 + b^2 + c^2)^3 - 3(a^3 + b^3 + c^3)^2 as much as possible.

coIinqiu Mar 11, 2025

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+95

Algebra

Let P = \log_8 3 and Q = \log_3 5. Express \log_{15} 72 in terms of P and Q. Your answer should no longer include any logarithms.

coIinqiu Mar 11, 2025

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+766

Algebra

Let x, y, and z be positive real numbers with x > y > z > 1 such that
\log_x y + \log_y z + \log_z x = \frac{13}{2},
\log_x z + \log_z y = 8.

Find \log_z x.

crimefightingvigiI Mar 11, 2025

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+766

Algebra

The equation
\sqrt[5]{7} x^{\log_7 x} = x^{\log_5 x}
has two positive roots a and b. Compute ab.

crimefightingvigiI Mar 11, 2025

Mar 10, 2025

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+1118

Algebra

Find \frac{a}{b} when 2 log(a - 2b) = log a + log (2b) - log(a + b).

AnswerscorrectIy Mar 10, 2025

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+1118

Algebra

If the two numbers \log_x 3 + \log_x 9 and \log_x 27 are equal, then what is their common value?

AnswerscorrectIy Mar 10, 2025

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+1118

Algebra

Given that log_{4n} 40 sqrt(3) = log_{2n} 10, find n^4.

AnswerscorrectIy Mar 10, 2025

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+766

Algebra

Let x, y, and z all exceed 1, and let w be a positive number such that \log_x w = 24, \log_x yx = 40, and \log_{xy^2} zw = 12. Find \log_z w.

crimefightingvigiI Mar 10, 2025

Mar 9, 2025

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+7

Two squares are inscribed in a semi-circle, as shown below. Find the radius of the semi-circle.

Two squares are inscribed in a semi-circle, as shown below. Find the radius of the semi-circle.

Tigergaming2464 Mar 9, 2025

Mar 8, 2025

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+766

Algebra

Let P = \log_8 3 and Q = \log_3 5. Express \log_{15} 72 in terms of P and Q. Your answer should no longer include any logarithms.

crimefightingvigiI Mar 8, 2025

0

1

0

+766

Algebra

Factor (ab + ac + bc)^3 - a^3 b^3 - a^3 c^3 - b^3 c^3 + (a^2 + b^2 + c^2)^3 - 3(a^3 + b^3 + c^3)^2 as much as possible.

crimefightingvigiI Mar 8, 2025

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+766

Algebra

Find all real numbers a and b such that
a + b = 14
a^3 + b^3 = 812 + a^2 + b^2

crimefightingvigiI Mar 8, 2025

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+75

Algebra

Find x if \log_2 x^2 + \log_{1/2} x + 3 \log_4 x = 5.

HumanBemg Mar 8, 2025

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+75

Algebra

Simplify 25^{\frac{1}{2} - \log 5 + \sqrt{3}}.

HumanBemg Mar 8, 2025

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+75

Algebra

Compute
\log_4 5 + \log_5 6 + \log_6 7+ \log_{2047} 2048 + \log_{2048} 2049

HumanBemg Mar 8, 2025

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+75

Algebra

Fill in the blanks, to make a true equation.
3/(3^2 - 1) + 3^2/(3^4 - 1) + 3^3/(3^6 - 1) + 3^4/(3^8 - 1) + ... + 3^(2(n - 1))/(3^(2n) - 1) = ___/___

HumanBemg Mar 8, 2025

Mar 7, 2025

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+75

Counting

Let S be the set {1, 2, 3, \dots, 10, 11, 12}. How many subsets of the set S have no two consecutive primes as members?

HumanBemg Mar 7, 2025

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+75

Algebra

Let
A_0 = 0
A_1 = 1
A_n = A_{n - 1} + A_{n - 2} for n ge 2
There is a unique ordered pair (c,d) such that c \alpha^n + d \beta^n is the closed form for sequence A_n. Find the ordered pair (c,d).

HumanBemg Mar 7, 2025

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+75

Algebra

Find the ordered pair (p,q) such that
F_n = p \alpha^n + q \beta^n.

HumanBemg Mar 7, 2025

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+75

Algebra

The Fibonacci sequence, is defined by F_0 = 0, F_1 = 1, and F_n = F_{n - 2} + F_{n - 1}. It turns out that
F_n = \frac{\alpha^n - \beta^n}{\sqrt{5}},
where \alpha = \frac{1 + \sqrt{5}}{2} and \beta = \frac{1 - \sqrt{5}}{2}.
The read more ..

HumanBemg Mar 7, 2025

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+75

Algebra

Find a closed form for
S_n = 1! \cdot (1^2 + 1) + 2! \cdot (2^2 + 2) + \dots + n! \cdot (n^2 + n).\]
for any integer n \ge 1. Your response should have a factorial.

HumanBemg Mar 7, 2025

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+75

Algebra

For a positive integer k, let
S_k = 1 \cdot 1! \cdot 2 + 2 \cdot 2! \cdot 3 + \dots + k \cdot k! \cdot (k + 1).
Find a closed form for S_k.

HumanBemg Mar 7, 2025

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