🧮 algebra

1. **Problem statement:** A rectangular piece of paper has sides of length $a$ cm and $b$ cm. It is cut in half vertically, creating two smaller rectangles each with dimensions $\f

1. **State the problem:** We have a vector $\mathbf{v} = \langle 12, 8 \rangle$ and a scalar $k$ such that $k\mathbf{v} = \langle 9, 6 \rangle$. We need to find the scalar $k$. 2.

1. **State the problem:** Solve the system of equations: $$x = 3y + 11$$

1. **State the problem:** Evaluate the product of two numbers in scientific notation: $$(7.7 \times 10^{-3})(3 \times 10^{-5})$$ 2. **Recall the exponential rule:** When multiplyin

1. **State the problem:** Multiply the numbers in scientific notation: $$(7.7 \times 10^{-3}) (3 \times 10^{-5})$$ 2. **Recall the multiplication rule for scientific notation:** Wh

1. **Stating the problem:** We are given population data over time and need to find an equation modeling this growth. 2. **Observing the data:** The population increases from 35.0

1. **State the problem:** Simplify the expression $$\frac{2x^4 y^3}{-9x^3 y^7}$$ and write the result using positive exponents. 2. **Recall the rules:**

1. **State the problem:** Simplify the expression $$\frac{2ab^{-1}}{2^{-1}a^{-4}b^2}$$ and write the result using positive exponents only. 2. **Recall the rules:**

1. **State the problem:** Simplify the expression $$\frac{5^5 z^{-5}}{5^6 z^{-8}}$$. 2. **Recall the laws of exponents:**

1. **State the problem:** Simplify the expression $$r \cdot r^{-8} \cdot r^{-4}$$ and write the result using positive exponents. 2. **Recall the exponent rule:** When multiplying p

1. **State the problem:** Simplify the expression $$\frac{p^8 p}{p^{-1}}$$ and write the result using positive exponents. 2. **Recall exponent rules:**

1. **State the problem:** Simplify the expression $$-8^0 - 2^0$$. 2. **Recall the rule:** Any nonzero number raised to the power of 0 is 1, i.e., $$a^0 = 1$$ for $$a \neq 0$$.

1. **State the problem:** Simplify the expression $$2^{-4} + 2^{-2}$$. 2. **Recall the rule for negative exponents:** $$a^{-n} = \frac{1}{a^n}$$ where $a \neq 0$ and $n$ is a posit

1. The problem is to simplify the expression $$\frac{y^{-6}}{y}$$. 2. Recall the rule for dividing powers with the same base: $$\frac{a^m}{a^n} = a^{m-n}$$.

1. **State the problem:** Simplify the expression $$\frac{y^{-6}}{y}$$. 2. **Recall the rule for dividing powers with the same base:** When dividing powers with the same base, subt

1. **State the problem:** Simplify the expression $$\frac{1}{p^{-1}}$$ and write the result using positive exponents. 2. **Recall the rule for negative exponents:** For any nonzero

1. We are asked to simplify the expression $$\left(-\frac{1}{4}\right)^{-3}$$. 2. Recall the rule for negative exponents: $$a^{-n} = \frac{1}{a^n}$$ where $a \neq 0$.

1. **State the problem:** Solve the system of equations: $$xy=4$$

1. **State the problem:** Solve the equation $$-7 + 7x - 3x = 37$$ for $$x$$. 2. **Combine like terms:** The terms with $$x$$ are $$7x$$ and $$-3x$$. Combine them:

1. The problem is to show that 180000 is greater than another number, which we will call $x$. 2. The inequality to prove is:

1. **State the problem:** Solve the system of equations: $$xy=4$$