Question

If \(a \Delta b=a^{b}+a b+\frac{a}{b}\), then \((9 \Delta 3)\) is equal to : (a) 729 (b) 927 (c) 759 (d) none of thesc

Short answer

Answer: The result of 9Δ3 is 759.

Step-by-step solution

Understand Given Operation Equation

We are given the operation equation as: \(a \Delta b = a^b + ab + \frac{a}{b}\).

Substitute Values of a and b

In order to find the result of \((9 \Delta 3)\), we need to substitute the values of \(a\) and \(b\). So, we will put \(a = 9\) and \(b = 3\) in the given equation.

Compute 9\Delta 3

Now, we calculate the value for \(9 \Delta 3\) by substituting the values into the equation: \(9 \Delta 3 = 9^3 + 9(3) + \frac{9}{3}\)

Evaluate the Expression

Next, we will solve the expression to get the result: \(9 \Delta 3 = 729 + 27 + 3\) \(9 \Delta 3 = 729 + 30\) \(9 \Delta 3 = 759\)

Choose the Correct Option

According to the calculation, option (c) is the correct option, as \((9 \Delta 3) = 759\).

Key concepts

Quantitative Aptitude

Quantitative aptitude is the ability to handle numerical and logical reasoning. It is essential for solving mathematical problems and is often evaluated in competitive exams and placement tests. To excel in quantitative aptitude, it is important to have a good grasp of basic mathematical operations as well as advanced concepts.

Students should focus on practicing a variety of problems to enhance their numerical reasoning and data interpretation skills. A problem like the one given in the textbook can be approached with a blend of mathematical knowledge and logical thought process, key ingredients for quantitative aptitude.

Exponentiation

Exponentiation is a mathematical operation that involves raising one number, the base, to the power of another number, the exponent. It's written as \(a^b\), where \(a\) is the base and \(b\) is the exponent. The exponent tells us how many times to multiply the base by itself.

Understanding Exponentiation

Knowing exponentiation is crucial to simplify and solve various mathematical problems. For instance, in the exercise \(9 \Delta 3\), the term \(9^3\) is the exponentiation part, which means \(9 \times 9 \times 9\), or \(729\).

Arithmetic Operations

Arithmetic operations are the foundation of mathematics. They include addition (+), subtraction (-), multiplication (\(\times\)), division (\(/\)), and exponentiation (\(^\)). Having a thorough understanding of these operations is crucial for solving a wide range of mathematical problems.

Application of Multiple Operations

Often, more than one arithmetic operation is combined in a single problem. This is evident in the exercise \(9 \Delta 3\), which combines exponentiation, multiplication, and division. The ability to accurately perform these operations in the correct order, following the BODMAS/BIDMAS rules, is essential for finding the correct solution.

Problem-Solving

Problem-solving in mathematics involves understanding the problem, devising a plan, carrying out the plan, and reviewing the solution. It's important to break down complex problems into simpler parts and tackle them step by step.

Steps in Problem-Solving

As illustrated in the given textbook solution, problem-solving is a sequential process. Beginning with understanding the unique operation defined in the problem, then substituting known values, and finally calculating step by step to arrive at the correct answer demonstrate this systematic approach. Strong problem-solving skills enable students to apply mathematical concepts to various scenarios beyond textbooks, such as real-life situations.