🧮 algebra

1. The problem is to find the value of $(\frac{2}{3})^{-1}$. 2. Recall the rule for negative exponents: For any nonzero number $a$ and integer $n$, $a^{-n} = \frac{1}{a^n}$.

1. The problem asks to write the expression $$\sqrt[3]{x^5}$$ as a rational exponent. 2. Recall the rule that a radical expression $$\sqrt[n]{x^m}$$ can be written as $$x^{\frac{m}

1. The problem is to evaluate the expression $$\sqrt[4]{81y^{8}x^{4}}$$. 2. Recall the rule for radicals: $$\sqrt[n]{a^{m}} = a^{\frac{m}{n}}$$. This means the fourth root of a pow

1. Problem (c): A dining room set is advertised with a 20% discount, and the discounted price is 75000. Calculate the original price before discount. 2. Formula: The discounted pri

1. Problem: Evaluate the function $f(x) = -3x + 4$ at $x=4$. 2. Formula: To find $f(a)$, substitute $x$ with $a$ in the function.

1. **State the problem:** Given the equation $$x^2 = 3^{2/3} + 3 - 2/3$$, prove that $$3x^2 + 9x = 8$$. 2. **Rewrite the given equation:**

1. The problem is to evaluate the expression $$(36x^4)^{\frac{1}{2}}$$. 2. We use the rule of exponents: $$(a^m)^n = a^{mn}$$ and the property of square roots: $$\sqrt{a} = a^{\fra

1. **State the problem:** Evaluate the expression $$ (36x^4)^{\frac{1}{2}} $$. 2. **Recall the rule:** The square root of a product is the product of the square roots, and the powe

1. The problem involves calculating the total price for a surface area given the dimensions and price per square meter. 2. First, calculate the area in square centimeters: $$\text{

1. The problem is to solve the first question from Class 7, Chapter 7, Exercise 7a, which typically involves algebraic expressions or equations. 2. Since the exact question is not

1. The problem is to find the value of $x$. 2. Since no equation or expression is given, we cannot solve for $x$ directly.

1. **State the problem:** A building is valued at 241000. Its value increases by 4.5%. We need to find the new value after the increase. 2. **Formula used:** To find the new value

1. **State the problem:** Solve the inequality $x + 5 > 7$. 2. **Formula and rules:** To solve inequalities, we isolate the variable on one side. Adding or subtracting the same num

1. Let's understand the expression $x=\pm \frac{2}{1}$. This means $x$ can be either positive or negative $\frac{2}{1}$.\n\n2. The symbol $\pm$ means "plus or minus," indicating tw

1. **State the problem:** Find the area of the region enclosed by the curves given by the function $y = (2x - 1)(2x + 1)$. 2. **Rewrite the function:** Expand the product to get a

1. **State the problem:** We want to find an explicit formula for the sequence $a_n$ defined by the recurrence relation $$a_n - 8a_{n-1} + 15a_{n-2} = n5^n, \quad n \geq 2$$

1. **State the problem:** Find the area of the region enclosed by the curve $y=(2x-1)(2x+1)$ and the x-axis. 2. **Rewrite the function:** Expand the product:

1. **State the problem:** We need to solve the system of inequalities:

1. **State the problem:** We are given two equations:

1. The problem asks to find the value of the sum $$1 + i + i^2 + i^3 + i^4 + \cdots + i^{16}$$ where $i$ is the imaginary unit with the property $i^2 = -1$. 2. Important property:

1. The problem involves a piecewise function: $$f(x) = \begin{cases} x^2 & \text{if } x > 10 \\ 3x + 1 & \text{if } x \leq 10 \end{cases}$$