🧮 algebra

1. **State the problem:** We have 100,000 picture frames divided into four types: Wooden, Metal, Stone, and Plastic. 2. **Define variables:** Let $P$ be the number of Plastic Frame

1. **Problem Statement:** Prove using mathematical induction that for any integer $n \geq 1$, $11^n - 6$ is divisible by 5.

1. **State the problem:** We need to find England's expected population size in 2012, given that it grows by a factor of 1.5 compared to 2011 and that in 2012 it is 12 times bigger

1. **Problem:** Solve the quadratic equation $$-x + (m + 1) x^2 - 4m = 0$$ 2. **Formula:** The general quadratic equation is $$ax^2 + bx + c = 0$$ and its solutions are given by th

1. **Problem Statement:** Calculate in percent how many more unemployed people are in Spain compared to France in 2011.

1. **Simplifier les radicaux suivants :** a) Simplifier $\sqrt[3]{\frac{18}{a} b^7}$

1. **Problem:** The product of three numbers is 3876. Two of the numbers are 17 and 19. Find the third number. 2. **Formula:** If the product of three numbers $a$, $b$, and $c$ is

1. **State the problem:** Sana cleaned $\frac{1}{4}$ of her house, and her sister cleaned $\frac{1}{5}$ of the remaining part. We need to find the ratio of the clean part to the un

1. The problem is not fully stated. Please provide the full details of Exercise 7 so I can assist you properly.

1. **Problem 6:** At a certain school, \(\frac{1}{4}\) of the teachers are women. Find the ratio of the number of male teachers to female teachers. - Total teachers = 1 (consider a

1. **State the problem:** We are given the function $f(t) = t^4 - 4t^2 + 11$ and want to analyze or simplify it. 2. **Formula and rules:** This is a polynomial function. We can try

1. **State the problem:** Solve the equation $\log_2 x + \log_4 x + \log_8 x = \frac{22}{3}$.\n\n2. **Recall the change of base formula:** For any logarithm, $\log_a b = \frac{\log

1. The problem is to evaluate the expression $\frac{5}{4} : 0.8$. 2. Division of a fraction by a number can be rewritten as multiplication by the reciprocal: $\frac{5}{4} : 0.8 = \

1. **Problem 1:** Find the ratio of red cars to blue cars given the ratios red:green = 5:6 and green:blue = 3:10. Step 1: Write the given ratios:

1. The problem is to use the Binomial Theorem to prove the expansion of $ (a+b)^n $. 2. The Binomial Theorem states that for any integer $ n \geq 0 $:

1. **Problem:** Calculate the value of the expression $12 - (-3) \times 4 \times (-20) - 6 \times 6 + (-6)$. 2. **Formula and rules:** Follow the order of operations (PEMDAS/BODMAS

1. **Problem Statement:** Write the equation of the parabola in conic form that opens to the right, has a focus of (2, -3), and a focal chord length of 32. 2. **Recall the formula

1. The problem is to factor completely the expression $$48x^{2n} - 75x^{4n}$$. 2. First, identify the greatest common factor (GCF) of the terms. Both terms have a factor of $$3x^{2

1. **Stating the problem:** Solve the logarithmic equation $$\log_5 5 - 20c + \log_3 2 = \log_5 20$$

1. **הבעיה:** נתונה סדרה הנדסית $A$ עם איברים שליליים $a_1, a_2, \ldots, a_m$ כאשר $m$ הוא מספר טבעי גדול מ-4. נתון שסכום $m$ האיברים הראשונים הוא פי 16 מסכום $m-4$ האיברים האחרוני

1. Let's clarify why we use absolute value in the 6th step of a problem involving square roots or logarithms. 2. Absolute value is used to ensure the result is non-negative because