🧮 algebra

1. **Problem:** Find the value of $$\frac{(a-b)^2+(b-c)^2}{(a-c)^2}$$ given that $a-b=b-c=2$. 2. **Formula and rule:** Since $a-c=(a-b)+(b-c)$, we first use the relation between th

1. Problem: factor $x^2-9$. 2. Use the difference of squares formula: $$a^2-b^2=(a-b)(a+b).$$

1. **State the problem:** Factor the expression $x^2 - 9$. 2. **Recall the formula:** This is a difference of squares, which follows the rule:

1. **State the problem:** Simplify the expression $$\sqrt[3]{\frac{49}{27}} \cdot j^{\frac{1}{3}}$$. 2. **Recall the property of radicals and exponents:** For any positive numbers

1. **Problem:** Simplify $\sqrt[3]{\frac{49}{27}}\,j^{1/3}$.\n2. Use the rule $a^{1/3}=\sqrt[3]{a}$. So $j^{1/3}=\sqrt[3]{j}$.\n3. Rewrite the expression as $\sqrt[3]{\frac{49}{27}

1. **State the problem:** Solve the system of linear equations for $x$ and $y$: $$\begin{cases} 2x + y = 8 \\ 2x - y = 12 \end{cases}$$

1. State the problem: Simplify the expression $\sqrt[3]{\frac{49}{27}}\, j^{1/3}$. 2. Use cube-root rules: For cube roots, $\sqrt[3]{a}\,\sqrt[3]{b}=\sqrt[3]{ab}$ and $\sqrt[3]{a}=

1. The problem asks to match each expression to its meaning among Total Cost, Amount of Tax, and Tax Rate. 2. The expressions given are:

1. State the problem: Find $\lim_{x\to 3} \dfrac{x^2-9}{x-3}$.\n\n2. Recall the key rule: If direct substitution gives an indeterminate form $0/0$, factor first.\n\n3. Factor the n

1. State the problem: Find $\sqrt[3]{64}$. 2. Use the cube root rule: $\sqrt[3]{a}=b$ means $b^3=a$.

1. **State the problem:** We want to find the population of San Diego given that it is about 85% of the population of Phoenix, which is 1.8 million people. 2. **Formula:** To find

1. **Problem:** Evaluate $\sqrt{64}$. 2. **Formula used:** The square root of a number is the value that, when multiplied by itself, gives the original number.

1. **Problem:** Find the value of $\sqrt{64}$. 2. **Formula:** The square root of a number is the value that, when multiplied by itself, gives the original number.

1. State the problem: Evaluate $3\sqrt{64}$. 2. Use the rule for perfect squares: $\sqrt{64}=8$ because $8^2=64$.

1. State the problem: Solve for $a$ in $a^3+a^2=36$.\n 2. Use the given equation: $a^3+a^2=36$.\n

1. **State the problem:** Solve the equation $$a^3 + a^2 = 36$$ for the variable $a$. 2. **Rewrite the equation:** We want to find $a$ such that $$a^3 + a^2 - 36 = 0$$.

1. **Problem:** Solve $a^3+a^2=36$ for $a$. 2. **Formula used:** First, rewrite the equation in standard form and factor when possible.

1. State the problem: Solve for $a$ in the equation $a^3+a^2=36$. 2. Factor the left side using the common factor $a^2$:

## Problem\nSolve for $a$ in the equation $a^3+a^2=36$.\n\n1. **Start with the given equation**\n$$a^3+a^2=36$$\n\n2. **Move everything to one side**\n$$a^3+a^2-36=0$$\n\n3. **Fact

1. El problema es simplificar la expresión algebraica dada o resolver la ecuación que tengas. 2. Para simplificar o resolver, primero identifica términos semejantes y aplica las pr

1. **State the problem:** Simplify the expression $10/2 \times 3 - 7$. 2. **Recall the order of operations (PEMDAS/BODMAS):**