🧮 algebra

1. **Problem Statement:** Evaluate without using a calculator:

1. **State the problem:** Given $\log_a y = k$, express each expression in terms of $k$. 2. **Recall logarithm rules:**

1. **State the problem:** We are given $\log_3 2 = 0.6309$ and $\log_3 5 = 1.465$ and need to evaluate:

1. **State the problem:** Express $$\log_c \sqrt{c} + \log_3 \sqrt{12}$$ as a single logarithm.

1. The problem asks if an expression is written as a single logarithm. 2. To express multiple logarithms as a single logarithm, use logarithm properties:

1. **State the problem:** Simplify the expression $$\log_c \sqrt{c} + \log_3 \sqrt{12}$$. 2. **Recall the logarithm and root rules:**

1. **State the problem:** Express the expression $$\frac{1}{2} \lg 25 - 3$$ as a single logarithm.

1. **State the problem:** Simplify the expression $$\log_6 \sqrt{3} + \log_6 \sqrt{24} - \log_6 \sqrt{2}$$. 2. **Recall logarithm properties:**

1. **Stating the problem:** Solve the equation $$88 + 2 = 3u^2 + 4$$ for the variable $$u$$.

1. **State the problem:** We want to find the effect on the graph of $f(x) = \frac{1}{x}$ when it is transformed to $g(x) = \frac{1}{x} + 13$. 2. **Recall the transformation rule:*

1. **State the problem:** Find the function $\left(\frac{f}{g}\right)(x)$ given $f(x) = \sqrt[3]{4x}$ and $g(x) = 3x + 5$, and determine the domain restrictions.

1. **State the problem:** Rationalize the denominator of the expression $$\frac{\sqrt{k}}{\sqrt{k} + \sqrt{g}}$$ where $k$ and $g$ are positive real numbers. 2. **Recall the formul

1. **State the problem:** Rationalize the denominator of the expression $$\frac{\sqrt{g}}{\sqrt{g} + \sqrt{k}}$$ where $g$ and $k$ are positive real numbers. 2. **Recall the formul

1. **State the problem:** Simplify the expression $$(5 + 2\sqrt{2})(3\sqrt{2} - 3)$$. 2. **Use the distributive property (FOIL) to expand:**

1. **State the problem:** Simplify the expression $14\sqrt{23} \cdot 5\sqrt{2}$. 2. **Recall the multiplication rule for radicals:** For any positive numbers $a$, $b$, $c$, and $d$

1. **State the problem:** Calculate the product of $3$ and the absolute value of $\frac{7}{2}$. 2. **Recall the absolute value rule:** The absolute value of a number is its distanc

1. **Stating the problem:** Solve the equation $$5^2 - x = \frac{1}{25}$$ for $$x$$. 2. **Recall the formula and rules:** We know $$5^2 = 25$$.

1. Problema 29: Fracțiile scrise de Andrei, Bianca și Corina. - Andrei scrie fracții de forma $\frac{a}{8}$ cu $a \mid 8$. Divizorii lui 8 sunt $1, 2, 4, 8$.

1. **State the problem:** We want to find how many children bought tickets to the fair given that adult tickets cost $25, children's tickets cost $10, 1000 people attended, and tot

1. Solve the system: $$-x = 4 - y$$

1. **State the problem:** Krystal has some $1 coins and $2 notes. The number of $2 notes is 4 times the number of $1 coins. The total value of all coins and notes is $54. We need t