🧮 algebra

1. **State the problem:** Given that $2^a=5$ and $5^b=7$, find the value of $2^{ab+1}$. 2. **Understand the problem:** We want to find $2^{ab+1}$. Notice that $ab+1 = ab + 1$, so w

1. **Problem Statement:** We want to find how many t-shirts and tickets must be sold to raise at least 5000 for charity.

1. **Problem statement:** We have a function $f$ defined for all real numbers such that $$f(x+y) = f\left(\frac{x}{2}\right) + f\left(\frac{y}{2}\right)$$ for all real $x,y$. We ar

1. The problem asks to add 9 to the square root of 3. 2. The square root of 3 is written as $\sqrt{3}$.

1. The problem is to simplify or work with an expression involving a square root without expanding it. 2. When dealing with square roots, the key rule is to keep the radical form i

1. **State the problem:** We need to simplify the expression $3\sqrt{6} + 9$. 2. **Recall the components:** The term $3\sqrt{6}$ is a product of 3 and the square root of 6, which i

1. **Énoncé du problème :** Calculer le déterminant de la matrice $A = \begin{pmatrix} 1 & 2 & 5 \\ 1 & 1 & 1 \\ 1 & -2 & -2 \end{pmatrix}$ et montrer qu'elle est inversible.

1. **State the problem:** Simplify the expression $$\sqrt[4]{2x^2} \cdot \sqrt[4]{8y^3}$$ and verify the given equality. 2. **Recall the property of radicals:** For the same root i

1. The problem is to solve the equation or expression given previously (not specified here, so assuming a generic solve request). 2. Since the original problem is not restated, let

1. **State the problem:** Simplify the expression $$2\pi^{1/2}(8 - 2t) \div 8\sqrt{\pi}$$. 2. **Recall the rules:**

1. The problem states that the points $(3,2)$ and $(-3,2)$ are maximum points, and the minimum value is $-6$ occurring at $x=0$. 2. We want to find a function $y=f(x)$ that satisfi

1. **State the problem:** Solve for $x$ in the equation $3x + 5 = 27$. 2. **Formula and rules:** To solve a linear equation, isolate the variable $x$ by performing inverse operatio

1. **State the problem:** Solve the linear equation $3x + 5 = 27$ for $x$. 2. **Formula and rules:** To solve for $x$, isolate the variable by performing inverse operations. Subtra

1. **State the problem:** Solve for $x$ in the equation $2(x + 4) = 16$. 2. **Use the distributive property:** Multiply 2 by each term inside the parentheses.

1. **Problem statement:** A contractor employs 20 men to finish a project in 50 days. After 15 days, 10 men leave, and the remaining 10 men increase their working hours from 6 hour

1. **State the problem:** Simplify the expression $$\frac{2}{1 - \sqrt{3}}$$. 2. **Formula and rule:** To simplify a fraction with a radical in the denominator, multiply numerator

1. The problem states that the sequence starts at 12 and each term increases by 6. 2. This is an arithmetic sequence where the first term $a_1 = 12$ and the common difference $d =

1. The problem is to find the first term in a Fibonacci-type sequence where the second term is 5 and the fifth term is 23. 2. Recall the Fibonacci-type sequence rule: each term is

1. The problem is to simplify the expression $2^{3} \times 4 \times 2^{-1}$. 2. Recall the rules of exponents: when multiplying powers with the same base, add the exponents: $a^{m}

1. **State the problem:** Solve the equation $16^x = 8^{2x+1}$ for $x$. 2. **Recall the formula and rules:** We can express both sides with the same base to compare exponents. Note

1. The problem is to solve the equation $\frac{1}{4} = 8 \times x$ for $x$. 2. The equation states that one quarter equals eight times some number $x$.