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Chapter 0: Problem 76

Simplify each expression. $$ \left(w^{n}-y^{m}\right)^{2} $$

Short Answer

Expert verified
The simplified expression is \( w^{2n} - 2w^{n}y^{m} + y^{2m} \).

Step by step solution

01

Identify the Given Expression

The given expression is \( \left(w^{n} - y^{m}\right)^{2} \). This is a binomial squared.
02

Apply the Binomial Square Formula

Recall that \( \left(a - b\right)^{2} = a^{2} - 2ab + b^{2} \). In this case, let \( a = w^{n} \) and \( b = y^{m} \).
03

Square the First Term

Calculate \( a^{2} = \left(w^{n}\right)^{2} = w^{2n} \).
04

Multiply the Two Terms and Double

Calculate \( 2ab = 2 \cdot w^{n} \cdot y^{m} = 2w^{n}y^{m} \).
05

Square the Second Term

Calculate \( b^{2} = \left(y^{m}\right)^{2} = y^{2m} \).
06

Combine All Parts

Putting it all together, the expression simplifies to \( w^{2n} - 2w^{n}y^{m} + y^{2m} \).

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Key Concepts

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

Simplification
Simplification is the process of making an algebraic expression easier to work with.
To simplify an expression, follow these steps:
  • First, identify the given expression.
  • Next, use algebraic rules or formulas to transform it.
In our example, we started with the expression\(\big(w^n - y^m\big)^2\). This form is a binomial squared. To simplify\(\big(w^n - y^m\big)^2\), we used the binomial square formula,\(\big(a-b\big)^2 = a^2 - 2ab + b^2\). We let\(a = w^n\) and\(b = y^m\). Following the steps, we transformed the given expression into a more manageable form.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators. Understanding parts of an algebraic expression can make the process of simplification easier.
  • Numbers are constants, like 2 or 5.
  • Variables are symbols that represent unknown values, like\(x\) or\(y\).
  • Operators are symbols that represent mathematical operations, like +, -, *, or /.
For the expression\(\big(w^n - y^m\big)^2\),\(w^n\)and\(y^m\)are both terms. The expression includes the operators - and ^ (exponentiation). Our goal was to simplify it using the binomial square formula. Each part of the algebraic expression played a role in our simplification steps.
Exponents
Exponents refer to the number of times a number (base) is multiplied by itself. In algebra, exponents make expressions more compact. For example,\(w^n\)indicates that\(w\)is multiplied by itself\(n\)times.
Here's a quick refresher on exponents:
  • \(a^m \cdot a^n = a^{m+n}\)
  • \(\big(a^m\big)^n = a^{mn}\)
  • \(a^0 = 1\)
In our simplification process, we worked with exponents like\(\big(w^n\big)^2\) to get\(w^{2n}\). By understanding and correctly applying the rules of exponents, we transformed the original expression into its simplified form. The expression \(\big(w^n - y^m\big)^2\) simplified to\(w^{2n} - 2w^n y^m + y^{2m}\), which utilizes the properties of exponents at each step.

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Most popular questions from this chapter

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