🧮 algebra

1. **State the problem:** We have a population model given by the formula $$P = 71.2e^{0.0329t}$$ where $t=1$ corresponds to the year 1971. We want to find the year when the popula

1. **State the problem:** We need to solve the equation or expression provided by the user. Since no specific problem was given, let's consider a general example: Solve for $x$ in

1. **State the problem:** Rewrite the radical expressions with positive exponents. 2. **Recall the rule:** A radical expression like $\sqrt[n]{x}$ can be written as $x^{\frac{1}{n}

1. The problem is to simplify the expression $$-14f^{\frac{1}{3}} b^5 \div \left(5 p^{\frac{1}{3}} y h\right)$$. 2. We start by writing the division as a fraction:

1. **State the problem:** Simplify the expression $2 - -2$. 2. **Understand the operation:** The expression contains a subtraction followed by a negative sign. Subtracting a negati

1. **State the problem:** A company produces 480 units in 8 hours. We need to find the production rate per hour. 2. **Formula:** Production rate per hour = \frac{\text{Total units

1. **State the problem:** We need to find the two possible values of $r$ that satisfy the equation $$\frac{216}{r^2} + 19 = 25.$$\n\n2. **Isolate the fraction:** Subtract 19 from b

1. The problem is to simplify an expression with a negative exponent. 2. Recall the rule for negative exponents: $$a^{-n} = \frac{1}{a^n}$$ where $a \neq 0$ and $n$ is a positive i

1. Solve the equation $8t = 56$. 2. To solve for $t$, divide both sides by 8:

1. The problem is to evaluate the expression $$(-8)^{\frac{2}{3}}$$. 2. Recall the rule for rational exponents: $$a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$$ whe

1. Stating the problem: Solve the equation $a x - 4 = 10$ for $x$. 2. Add 4 to both sides to isolate the term with $x$:

1. **Problem Statement:** Find the inverse function $f^{-1}$ of $f(x) = 3x$, check the answer, and find the domain and range of both $f$ and $f^{-1}$. Also, graph $f$, $f^{-1}$, an

1. **State the problem:** Factorize the quadratic expression $2x^2 + 6x + 4$. 2. **Recall the factoring formula:** For a quadratic $ax^2 + bx + c$, we look for two numbers that mul

1. **State the problem:** Simplify the expression $$\frac{-3 + (-2 - 9) - 1}{(-5)(-2 - 3 + 6)}$$. 2. **Simplify the numerator:**

1. **State the problem:** Mary sells $28,000 worth of furniture and has already received a draw of $766.15. We need to calculate how much commission is still owed to her based on t

1. **State the problem:** Simplify the expression $$9q^2 + 6q^2 + 12p^3 + 3p^3 + 18q^2$$. 2. **Identify like terms:** Terms with the same variable and exponent can be combined.

1. The problem asks to find $(f - g)(x)$ given $f(x) = 7x - 6$ and $g(x) = 2x - 4$. 2. The formula for the difference of two functions is:

1. **State the problem:** Pamela Mello earns a commission of 2.6% on the first 70000 and 3.4% on any sales above 70000. Her weekly sales are 86800. We need to find her total commis

1. **Stating the problem:** We want to find a function that models the number of students participating in sports after $x$ years, given that the initial number is 317 and it incre

1. **State the problem:** A customer buys an electronic device for 650. Each year, its value decreases by 39%. We want to determine which statement about the graph of the device's

1. **State the problem:** Determine which statement about the graph of the function $y = 16(0.5)^x$ is NOT true. 2. **Recall the function and its properties:** The function is an e