Problem 104 \(\left(x-y^{-2}\right)^{3}=\fra... [FREE SOLUTION]

Chapter 5: Problem 104

\(\left(x-y^{-2}\right)^{3}=\frac{1}{x^{9} y^{6}} \quad\)

Short Answer

Expert verified

After simplifying the expression and comparing both sides, we observe that the two sides do not equate to each other as one might have expected.

Step by step solution

Simplify the Left Side

We will make use of the binomial theorem to solve the left side of the formula: \( (x-y^{-2})^{3} \). The binomial theorem tells us that \( (a+b)^{3} = a^{3} + 3a^{2}b + 3ab^{2} + b^{3} \). So using this, we get: \((x-y^{-2})^{3} = x^{3} - 3x^{2}y^{-2} + 3xy^{-4} - y^{-6}\).

Compare with the Right Side

Now let's compare it with the reciprocal of the right-hand side which is \( x^{9}y^{6} \). Equating them, we have: \( x^{3} - 3x^{2}y^{-2} + 3xy^{-4} - y^{-6} = x^{9}y^{6} \).

Conclude

After equating the expanded form of the left-hand side to the right hand side ( in reciprocal form ), we observe that there is no common ground. Hence little further simplification or solution can be derived from the comparison.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponent Laws

Exponent laws are the rules that govern the operations and manipulations of exponents in mathematical expressions. They allow us to simplify expressions involving powers of a number or a variable. Understanding exponent laws is crucial when working with expressions like \((x - y^{-2})^3\).

Some key exponent laws include:

Product of Powers: If you multiply like bases, you add the exponents. For example, \(a^m \times a^n = a^{m+n}\).
Quotient of Powers: If you divide like bases, you subtract the exponents. For example, \(\frac{a^m}{a^n} = a^{m-n}\).
Power of a Power: To raise a power to another power, you multiply the exponents. For instance, \((a^m)^n = a^{mn}\).
Negative Exponents: A negative exponent means you take the reciprocal of the base. For example, \(a^{-n} = \frac{1}{a^n}\).

Using these laws, you can begin to simplify complex expressions, such as transforming \(y^{-2}\) into \(\frac{1}{y^2}\), which can help in polynomial expansions.

Polynomial Expansion

Polynomial expansion is a technique used to express a power of a binomial, such as \((x - y^{-2})^3\), as a polynomial. This is where the binomial theorem comes into play, making the process systematic and efficient.

The binomial theorem states that for any positive integer \(n\), the expression \((a + b)^n\) can be expanded into a sum involving terms of the form \( \binom{n}{k} a^{n-k} b^k\) where \(k\) ranges from 0 to \(n\). In simpler terms, it breaks down the binomial raised to a power into manageable polynomial terms:

Consider \((x - y^{-2})^3\), using the binomial theorem, you get:
\(x^3 - 3x^2y^{-2} + 3xy^{-4} - y^{-6}\).
For each term, \(a\) and \(b\) from the binomial are raised to powers that add up to the total power \(n\).

This method not only simplifies the calculation but also helps in understanding the structure of polynomial equations in algebra.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They serve as the building blocks of algebraic equations and are essential in solving mathematical problems.

Understanding algebraic expressions is critical when dealing with problems like \((x - y^{-2})^3\). This expression is a combination of the variable \(x\), a negative exponent variable \(y^{-2}\), and the operation of taking the power.

Key aspects of algebraic expressions include:

Terms: Parts of the expression separated by addition or subtraction signs. For instance, in \(x^3 - 3x^2y^{-2} + 3xy^{-4} - y^{-6}\), each section separated by \(-\) or \(+\) signs is a term.
Coefficients: Numbers that multiply the variables, such as 3 in \(-3x^2y^{-2}\).
Constants: Numbers without any variables. In some expressions, variables will cancel out resulting in a constant.

By breaking down and understanding each part, students can better grasp how to manipulate and solve algebraic expressions, leading to solutions in exercises involving polynomial expansions.

91影视

\(\left(x-y^{-2}\right)^{3}=\frac{1}{x^{9} y^{6}} \quad\)

Short Answer

Step by step solution

Simplify the Left Side

Compare with the Right Side

Conclude

Key Concepts

Exponent Laws

Polynomial Expansion

Algebraic Expressions

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