🧮 algebra

1. We need to expand the expression $ (4x + 5y)^2 $. 2. Recall the identity for the square of a binomial: $ (a+b)^2 = a^2 + 2ab + b^2 $.

1. We need to find the smallest term of the expression $$2n + \frac{512}{n^2}$$ where $n$ is a positive real number. 2. To find the minimum value, we take the derivative of the exp

1. **State the problem:** Express the numbers 16 and 14 as powers or combinations of base 2. 2. **Express 16 with base 2:**

1. We are asked to determine if the given statements are TRUE or FALSE and justify the answers. (i) Distance between points $(3x, -3x)$ and $(-x, -6x)$ is $-5x$ if $x<0$.

1. We are asked to factorize the quadratic expression $6t^2 - 7t - 20$. 2. First, multiply the leading coefficient and the constant term: $6 \times (-20) = -120$.

1. The problem is to find the smallest value of the expression $3n^2 - 10n - 14$ for integer values of $n$. 2. This is a quadratic expression in standard form $an^2 + bn + c$ with

1. State the problem: Solve the equation $$\frac{7x - 6}{x - 18} = -\frac{2}{3}$$ for $x$. 2. Cross-multiply to eliminate the fractions:

1. **Problem Statement:** Find the smallest term of the expression $$\frac{2^2}{n^2}$$ where $n$ is a variable. 2. **Rewrite the expression:** The expression simplifies as $$\frac{

1. The problem involves simplifying or evaluating the expression for "3 multiple (-2 -1)", which can be interpreted as multiplying 3 by the sum of -2 and -1. 2. Calculate the sum i

1. The problem is to calculate $16$ divided by $-2$, and then divide the result by $2$. 2. First, divide $16$ by $-2$:

1. The problem is to divide 0.8 by -2. 2. Write the expression as a fraction: $$\frac{0.8}{-2}$$.

1. Stating the problem: Calculate the value of $\frac{-1000}{-10}$.\n\n2. Simplify the division: Dividing a negative number by another negative number results in a positive number.

1. **State the problem:** Write each expression as a single power of 4. 2. **Recall:**

1. **State the problem:** We need to find the smallest value of the expression $$n + \frac{100}{n}$$ where $n$ is a positive real number. 2. **Rewrite the expression:** The functio

1. The problem is to find the smallest value (minimum) of the quadratic expression $$n^2 - 5n + 1$$. 2. A quadratic expression in the form $$an^2 + bn + c$$ with $$a > 0$$ opens up

1. Stating the problem: Simplify the expression $a^{\log_a b}$.\n\n2. Recall the logarithm power rule: For any positive $a \neq 1$ and positive $b$, $a^{\log_a b} = b$. This is bec

1. **Problem 1:** Given $x = \sqrt{7} - \sqrt{5}$, express $\frac{1}{x}$ in the form $\sqrt{a} + \sqrt{b}$. Step 1: Start with $\frac{1}{x} = \frac{1}{\sqrt{7} - \sqrt{5}}$.

1. Stating the problem: We want to find the greatest limit of the function $$f(n) = \frac{\sqrt{n}}{9 + n}$$ as $$n$$ approaches infinity or other relevant points. 2. Consider the

1. State the problem: Solve the quadratic equation $$3x^2 + 10x - 8 = 0$$. 2. Calculate the discriminant: $$\Delta = b^2 - 4ac = 10^2 - 4 \times 3 \times (-8) = 100 + 96 = 196$$.

1. Stating the problem: Solve the quadratic equation $x^2 - 4 = 0$. 2. Add 4 to both sides to isolate the square term:

1. The problem is to find the missing number represented by "?" in the sequence: 8 15 6, 4 7 6, 2 5 9, 15 ? 4. 2. Let's examine each group of three numbers to identify a pattern.