Problem 28 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\alpha=18^{3}+7^{3}+3.18 .7 .25\), and \(\beta=3^{6}+6.243 .2+15.81 .4+20.27 .8\) \(+15.9 .16+6.3 .32+64\) then the value of \(\alpha \beta^{-1}\) is (A) 1 (B) 5 (C) 25 (D) 100

Step-by-Step Solution

Answer

The value of \(\alpha \beta^{-1}\) is 1 (Option A).

Step 1: Simplify expression for alpha

Start by simplifying \(\alpha = 18^3 + 7^3 + 3 \times 18 \times 7 \times 25\). First, calculate \(18 \times 7\ = 126\), then \(3 \times 126 = 378\), and finally, \(378 \times 25 = 9450\).Now calculating the cubes: \[18^3 = 5832\] \[7^3 = 343\]Substitute back the values:\[\alpha = 5832 + 343 + 9450 = 15625\].

Step 2: Simplify expression for beta

Now simplify \(\beta = 3^6 + 6 \times 243 \times 2 + 15 \times 81 \times 4 + 20 \times 27 \times 8 + 15 \times 9 \times 16 + 6 \times 3 \times 32 + 64\).First, compute the base calculations:\[3^6 = 729\]Then calculate each multiplication separately:\[6 \times 243 \times 2 = 2916\]\[15 \times 81 \times 4 = 4860\]\[20 \times 27 \times 8 = 4320\]\[15 \times 9 \times 16 = 2160\]\[6 \times 3 \times 32 = 576\]Now, add these values together, along with 64:\[\beta = 729 + 2916 + 4860 + 4320 + 2160 + 576 + 64\].

Step 3: Calculate beta

We need to sum up the calculated values of \(\beta\):\[\beta = 729 + 2916 + 4860 + 4320 + 2160 + 576 + 64 = 15625\].

Step 4: Calculate alpha beta inverse

Now compute \(\alpha \beta^{-1}\) using the values of \(\alpha\) and \(\beta\) we determined:Both \(\alpha\) and \(\beta\) are 15625, so: \[\alpha \beta^{-1} = \frac{15625}{15625} = 1\].

Key Concepts

Algebraic Expressions

The fundamental building blocks of mathematics, algebraic expressions, are combinations of numbers, variables, and operators like addition and multiplication. They form the backbone of many mathematical problems. In this exercise, we explore expressions for \( \alpha \) and \( \beta \), each consisting of various terms that require simplification.

• Components of an Expression: An algebraic expression can have constant terms (like numbers alone), variables (like \( x, y \) representing unknowns), and coefficients (numbers multiplied by variables).


• Operations: Expressions often involve operations such as addition, subtraction, multiplication, and division. These operations must follow the order of operations (PEMDAS/BODMAS).


• Like Terms: When terms in the expression have the same variable components, they can be combined. For example, \( 3x + 4x = 7x \).

In simplifying \( \alpha \) and \( \beta \), each sub-expression must be evaluated using these principles to ensure an accurate solution. Clear recognition and manipulation of these expressions lead to a smooth simplification process.

Solving Equations

To find the solution to an equation, our goal is to determine the value of the unknown that satisfies the equation. In this exercise, we use computations to solve for \( \alpha \beta^{-1} \). To do that, let's focus on how we tackle solving equations:

• Basic Approach: An equation asserts that two expressions are equal. For straightforward equations like \( x + 3 = 5 \), subtract 3 from both sides to solve for \( x \).


• Employing Inverses: If you encounter multiplication or division, use the inverse operation to isolate the variable. Thus, in this exercise, \( \alpha \beta^{-1} \) is essentially solving the division of \( \alpha \) by \( \beta \).


• Ensuring Balance: Whatever you do on one side of the equation, you must do to the other. This maintains equality.

Understanding these basic principles makes handling algebraic expressions within an equation much more approachable. By applying consistent operations to both sides, we arrived at the solution for \( \alpha \beta^{-1} \) equaling 1.

Mathematical Simplification

Simplification in mathematics is about making expressions and equations easier to work with by consolidating complex parts into simpler forms. Simplification helps in understanding and solving problems more efficiently. Here's how it applies to this problem:

• Simplifying Terms: Break down complex multiplications or powers into their simplest forms. For example, calculating the powers \( 18^3 \) and \( 3^6 \), and multiplying numbers one by one.


• Combining Like Terms: Add or subtract similar terms. Here, each calculated term in \( \alpha \) and \( \beta \) was added together after simplifying.


• Validate Prior Results: To confirm correctness, re-calculate slightly different paths or double-check using different methods, if needed.

Through simplification, each part of the expressions for \( \alpha \) and \( \beta \) reduced down to single numbers, 15625, showing the power of simplification in reaching a conclusion efficiently.