🧮 algebra

1. Start with the expression: $3\sqrt{2} + 2\sqrt{18} - 5\sqrt{50}$. 2. Simplify each square root term:

1. **State the problem:** We need to find five consecutive even integers whose sum is 470. 2. **Define variables:** Let the smallest even integer be $x$.

1. **State the problem:** Jack can carry a pail of water uphill alone in 40 minutes, Jill can do it alone in 25 minutes, and we want to find how long it takes if they help each oth

1. State the problem: Solve for $x$ in the equation $\log_2(4x - 12) = 5$.\n\n2. Recall the definition of logarithm: $\log_b(a) = c$ means $b^c = a$.\n\n3. Applying this to our equ

1. **State the problem:** Kevin can clean a room alone in 36 minutes. When Ben helps, it takes 9 minutes together. We need to find how long it takes Ben to clean alone. 2. **Define

1. **State the problem:** We need to find the smallest of four consecutive multiples of 7 whose sum is 1582. 2. **Define variables:** Let the smallest multiple be $7n$, where $n$ i

1. The problem states we have a geometric progression (GP) starting with 2, 6, 18, 54, ... and we want to find how many terms are added to get a sum of 19682. 2. Identify the first

1. **State the problem:** Find the sum of the first seven terms of an arithmetic progression (AP) where the first four terms are -7, -3, 1, and 5. 2. **Find the common difference $

1. The problem states that 𝑎 and 𝑏 are both odd numbers. 2. Recall that an odd number can be written as $2k+1$ for some integer $k$.

1. State the problem: We want to find the 20th term ($a_{20}$) of an arithmetic sequence. 2. Recall the formula for the $n$-th term in an arithmetic sequence:

1. Problem: For each pair of functions, identify one characteristic they have in common and one that distinguishes them. 2. a) Functions: $f(x) = \frac{1}{x}$ and $g(x) = x$

1. **State the problem:** We need to divide the expression $12x^7y^3 + 8xy^4 - 24x^8y^9$ by $2xty^2$. 2. **Write the division:**

1. **State the problem:** We need to find the first term $a_1$ of an arithmetic sequence given that the 19th term $a_{19} = 417$ and the 12th term $a_{12} = 319$.

1. **State the problem:** Multiply the expressions $ (3x + 2y) $ and $ (4x - 3y) $. 2. **Apply the distributive property:** Multiply each term in the first parenthesis by each term

1. Stating the problem: We need to evaluate the expression $$\sqrt{(-2\sqrt{3} - 2\sqrt{3})^2}$$. 2. Simplify inside the parentheses: Combine like terms.

1. **Problem:** Simplify the square root of 98 by removing all perfect squares from inside the root. 2. First, find the factors of 98 and identify any perfect squares.

1. We are asked to simplify the square root of 450, i.e., simplify $$\sqrt{450}$$. 2. To simplify a square root, factor the number inside into prime factors or perfect squares.

1. **State the problem:** Evaluate $$324^{\frac{1}{4}} \cdot \sqrt[4]{\frac{1}{4}}$$. 2. **Rewrite the expression using fractional exponents:** $$324^{\frac{1}{4}} \cdot \left(\fra

1. We are asked to simplify the expression $\left(2^{-7} \cdot 5^5\right)^2$. 2. Apply the power of a product rule: $\left(a \cdot b\right)^n = a^n \cdot b^n$. So,

1. State the problem: Simplify the expression $$\frac{-8 \cdot 10^{7}}{-4 \cdot 10^{6}}$$. 2. Separate the constants and the powers of 10:

1. **State the problem:** Simplify the expression $$\frac{-8 \cdot 10^7}{-4 \cdot 10^6}$$. 2. **Simplify the fraction:** First, divide the coefficients: