🧮 algebra

1. The problem asks to find the sum of the arithmetic progression (AP) series starting from 80, 81, 82, ..., up to 139, 140 which are 100 positive integers. 2. First, let's identif

1. Stated problem: Simplify the expression $$(-d^5)^4$$. 2. Apply the power of a power rule: $$(a^m)^n = a^{m \times n}$$, so $$(-d^5)^4 = (-1)^4 \times (d^5)^4$$.

1. **State the problem:** We want to verify and complete the square for the quadratic expression $4x^2 + 14x$. 2. **Rewrite the expression:** Write $4x^2 + 14x$ as $4x^2 + 4x + 8$

1. Stating the problem: Simplify the expression $(mn^2)^4$. 2. Apply the power to each factor inside the parentheses: $(mn^2)^4 = m^4 (n^2)^4$.

1. **State the problem:** We have 12 bags of apples with a mean of 8 apples per bag. The frequencies of apples in 11 bags are given. We need to find the number of apples in the 12t

1. The given expression is $\frac{mn}{2}\times 4$. 2. First, rewrite the multiplication: $$\frac{mn}{2} \times 4 = \frac{mn}{2} \times \frac{4}{1}.$$

1. We are given 12 bags of apples and the mean number of apples in each bag is 8. 2. The total number of apples in all 12 bags combined is calculated by multiplying the mean by the

1. আমরা সমস্যাটি আলোচনা করি: আমরা $-17 + 6i \sqrt{2}$ এই জটিল সংখ্যা সংক্রান্ত কোন প্রশ্ন বুঝতে চাই। 2. এখানে $i$ হলো কাল্পনিক একক, যার মান $i^2 = -1$।

1. প্রদত্ত সংখ্যাটি হলো একটি জটিল সংখ্যা যা $-17 + 6i\sqrt{2}$। এখানে $-17$ হলো বাস্তব অংশ এবং $6i\sqrt{2}$ হলো কাল্পনিক অংশ। 2. আমরা এর মডুলাস (মডিউল) বা দৈর্ঘ্য এবং আর্গুমেন্ট (ক

1. Let's start by calculating each symbol value step-by-step. 2. For $$\Omega$$:

1. The problem asks to round the number 7.583 to 2 decimal places. 2. Identify the digit at the 2nd decimal place which is the hundredths place. Here, it is 8 in 7.583.

1. The problem is to fully simplify the expression $2\sqrt{112}$. 2. Factorize the number under the square root to find perfect squares:

1. Stating the problem: We want to calculate $2\sqrt{3} \times \sqrt{5}$ and write the result in the form $\sqrt{q}$, where $q$ is an integer. 2. Multiply the expressions:

1. Given the expression $(2n^4)^{-5}$, we want to simplify it. 2. Apply the power of a power rule: $(a^m)^n = a^{m \cdot n}$. Here, $a = 2n^4$, $m=1$, and $n=-5$.

1. **State the problem:** Solve the system of linear equations:

1. The user asks if a certain problem can be solved by rationalization, factorization, or L'Hopital's Rule. 2. Without a specific problem statement, it's important to clarify how e

1. The problem gives the sequence: 1, 8, 27, 64, 125, ... 2. Identify the pattern: these are cubes of natural numbers.

1. The problem is to find the next three terms and the rule for the sequence: 1, 4, 9, 16, 25, ... 2. Observe the pattern of the terms: 1 = $1^2$, 4 = $2^2$, 9 = $3^2$, 16 = $4^2$,

1. **State the problem:** Solve the system of equations using the elimination method: $$\begin{cases}-x - 5y - 5z = 2\\4x - 5y + 4z = 19\\x + 5y - z = -20\end{cases}$$

1. State the problem: Solve the equation $3x + 5 = 7$ for $x$. 2. Subtract 5 from both sides to isolate the term with $x$:

1. Problem: For each sequence, find the next three terms and the rule for the terms. 2. Sequence a: $1, 2, 4, 8, 16, 32, \dots$