🧮 algebra

1. **State the problem:** Simplify the expression $\frac{1}{x} + 2 - \frac{1}{x}$. 2. **Identify like terms:** Notice that $\frac{1}{x}$ and $-\frac{1}{x}$ are additive inverses.

1. **State the problem:** We have multiple lawn sections with given algebraic dimensions and know their areas are equal. We need to find the dimensions of each section. 2. **Identi

1. **Problem statement:** Given that $a > b$ and both $a$ and $b$ are positive numbers, prove that for any natural number $n$, the inequality $\frac{n}{a} < \frac{n}{b}$ holds. 2.

1. The problem states: If $7^m = t$, find an expression for $7^{m+3}$ in terms of $t$. 2. Recall the exponent rule: $a^{x+y} = a^x \times a^y$.

1. **State the problem:** Simplify the expression $$-\sqrt[3]{2x^{2}y^{2}} \cdot 2\sqrt[3]{15x^{5}y}$$. 2. **Recall the property of cube roots:** The product of cube roots can be c

1. The problem is to simplify the expression $6 \sqrt{13}$. 2. Recall that $\sqrt{a}$ represents the square root of $a$, and it cannot be simplified further if $a$ is not a perfect

1. **State the problem:** Simplify the expression $$\sqrt[5]{32y^{25}}$$. 2. **Recall the formula:** The fifth root of a product is the product of the fifth roots: $$\sqrt[5]{ab} =

1. The problem is to simplify the expression $14) -\sqrt{81c^{48}d^{64}}$. 2. First, note that the square root of a product is the product of the square roots: $$\sqrt{81c^{48}d^{6

1. The problem asks to write the given numbers as powers of ten. 2. Recall that any number can be written as $10^n$ where $n$ is the exponent representing how many times 10 is mult

1. **Problem:** Calculate $$\left(-\frac{8}{15}\right) \times \left(7 \frac{1}{4} - 3 \frac{1}{2}\right)$$. 2. **Convert mixed numbers to improper fractions:**

1. **State the problem:** Simplify the expression $$(3 + 6n^5 - 8n^4) - (-6n^4 - 3n + 8n^5).$$ 2. **Recall the rule:** When subtracting a polynomial, distribute the minus sign to e

1. **State the problem:** Simplify the expression $$(4n - 3n^3) - (3n^3 + 4n)$$. 2. **Apply the distributive property:** Remove the parentheses by distributing the minus sign to th

1. **State the problem:** Simplify the expression $$(4r^3 + 3r^4) - (r^4 - 5r^3).$$ 2. **Apply the distributive property:** Remove the parentheses by distributing the minus sign to

1. The problem is to simplify the expression $$(3a^2 + 1) - (4 + 2a^2)$$. 2. Use the distributive property to remove the parentheses: $$3a^2 + 1 - 4 - 2a^2$$.

1. **State the problem:** We want to find the population of Bigtimeville in the year 2008 given that in 2025 the population is 63,881 and the net growth rate is $\frac{8}{1000}$ pe

1. The problem asks to evaluate $|(-16) + (-4)|^2$. 2. First, simplify inside the absolute value: $(-16) + (-4) = -20$.

1. The problem asks to evaluate $\left(\frac{36}{6}\right)^3$. 2. First, simplify the fraction inside the parentheses: $\frac{36}{6} = 6$.

1. **State the problem:** Simplify and solve the equation $$\frac{3x - 1}{6x - 3} + \frac{1}{4x^2 - 1} = \frac{-x}{2x + 1}.$$\n\n2. **Identify and factor denominators:**\n- $6x - 3

1. **State the problem:** Simplify the expression $$2 - \frac{3x+2}{x+2} - \frac{2}{7}(2-x)$$. 2. **Rewrite the expression:**

1. **State the problem:** Solve the equation $$(4x+1)(x+3)-3(x-2)=(2x-3)^2$$ for $x$. 2. **Expand both sides:**

1. **State the problem:** Find the inverse of the function $$f(x) = (x - 3)^2 + 3$$ for $$x \geq 3$$ and verify the result. 2. **Recall the definition of inverse function:** The in