Problem 159 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ y=(x+1)^{\frac{2}{2}} $$

Step-by-Step Solution

Answer

The simplified expression for y is \(y = x + 1\).

Step 1: Simplify the exponent

First, simplify the exponent by dividing the numerator (2) by the denominator (2). This results in: \[ y = (x + 1)^{(\frac{2}{2})} \Rightarrow y = (x + 1)^{1} \]

Step 2: Simplify the expression further

As raising a number to the power of 1 is simply the number itself, the expression simplifies to: \[ y = (x + 1)^{1} \Rightarrow y = x + 1 \] The simplified expression for y is y = x + 1.

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. These expressions represent quantities without fixed values, often involving unknowns like x and y. Simplifying algebraic expressions is a common task in algebra that involves reducing the expressions to their simplest form. This can include combining like terms, using distribution, and simplifying exponents, which leads us to the power of understanding exponent rules.

Exponent Rules

Exponents, also known as powers, are shorthand for repeated multiplication. For instance, when we write \( x^2 \), we mean \( x \) times \( x \). When simplifying exponents, several key rules come into play:

• The Product Rule: \( x^a \cdot x^b = x^{a+b} \)
• The Quotient Rule: \( \frac{x^a}{x^b} = x^{a-b} \)
• The Power Rule: \( (x^a)^b = x^{a\cdot b} \)
• Zero Exponent Rule: \( x^0 = 1 \) (if \( x eq 0 \))
• Negative Exponent Rule: \( x^{-a} = \frac{1}{x^a} \)

Another important concept is simplifying fractional exponents, where the numerator represents the power and the denominator represents the root. For example, in the original exercise, the expression \( (x+1)^{\frac{2}{2}} \) can be simplified by applying such exponent rules.

Solving Algebraic Equations

Solving algebraic equations is the process of finding the value(s) of the variable(s) that make the equation true. It involves moving terms from one side of the equation to the other and simplifying the equation step by step until the variable is isolated. Understanding the properties of equality and operations on algebraic expressions is crucial. The foundational steps include simplifying expressions on each side of the equation, collecting like terms, and following an order of operations to ensure accuracy. It's essential to work through each equation methodically, often resulting in finding the solution or solutions for the variable. Remember, the ultimate goal is to get the variable on one side of the equation by itself, clearly showing what it equals.