Problem 417 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Find all pairs \((x, y)\) of real numbers such that \(16^{x^{2}+y}+16^{x+y^{2}}=1\).

Step-by-Step Solution

Answer

The solutions are the pairs \((x, y) = (0, -1)\) and \((x, y) = (-1, 0)\).

Step 1: Recognize Properties of Exponential Functions

Observe that \(16=2^{4}\) and apply the property of exponential functions \(b^{m+n}=b^m*b^n\) so find that the equation can be rewritten as \( (2^{4})^{x^2+y} + (2^{4})^{x+y^2} = 1\). Which simplifies to \(2^{4x^2+4y} + 2^{4x+4y^2} = 1\).

Step 2: Rearrange The Equation

Rearrange the equation by subtracting \(2^{4x+4y^2}\) from both sides, to get \(2^{4x^2+4y} = 1 -2^{4x+4y^2}\).

Step 3: Analyze Possible Solutions

Observe that for this equation to hold, both \(4x^2+4y\) and \(4x+4y^2\) must be negative (since the left-hand side is positive but less than 1 and the right-hand side is positive but less than 1). There are only a few pairs \((x, y)\) that satisfy these two inequalities - namely \((x,y) = (0,-1)\), and \((x,y) = (-1,0)\). These are the solutions.

Key Concepts

Real Numbers

Real numbers include all the numbers we deal with in algebra: whole numbers, fractions, and irrational numbers.
The set of real numbers is vast and continuous, filling in gaps along the number line.
When solving equations, identifying real number solutions means finding specific value pairs that hold true for an equation.

• Real numbers can be positive, negative, or zero.
• In this problem, both variables \(x\) and \(y\) need to be real numbers.
• They can take on any decimal or fractional value.

This flexibility allows us to explore solutions without limiting ourselves to specific types like integers or whole numbers.

Properties of Exponents

Exponents are powerful tools in algebra, and they provide a shorthand way to represent repeated multiplication.
Key properties of exponents help us simplify and solve equations efficiently.

• The property \(b^{m+n} = b^m \cdot b^n\) allows us to break down complicated expressions.
• In this problem, we recognize that \(16 = 2^4\), which allows rewriting the equation using base 2 for easier manipulation.
• Understanding how to manipulate exponents is crucial for reducing equations into more manageable forms.

By applying these properties, you can transform complex exponential equations into simpler forms for analysis.

Equation Solving

Solving equations involves finding the values of variables that make an equation true.
This process can include manipulating the equation to isolate variables or using logical reasoning.

• In this case, rearranging terms helps isolate parts of the equation to explore potential solutions.
• The equation can be rewritten into a form that is easier to analyze or solve.
• Once simplified, we attempt to find values of \((x, y)\) that satisfy both sides of the equation.

Understanding how to systematically rearrange and manipulate equations is essential for finding solutions efficiently.

Inequalities in Equations

Inequalities extend the concept of equations to explore ranges of solutions.
They help in understanding not just strict equalities but conditions that must hold true under certain constraints.

• For \(2^{4x^2+4y} + 2^{4x+4y^2} = 1\), it is observed that both expressions need to be less than 1.
• This leads us to examine when these functions become smaller than 1 by looking at the exponential terms.
• Both \(4x^2+4y\) and \(4x+4y^2\) need to be negative for the exponential results to be in the required range.

Solving these inequalities gives insight into which pairs \((x, y)\) are valid, resulting in the solutions \((0, -1)\) and \((-1, 0)\). Understanding these concepts helps us identify all possible real-number solutions.