Question

Simplify the expression.\((-2 w)^{5}\)

Short answer

The simplified form of the expression \((-2 w)^{5}\) is \( -32 * w^5 \).

Step-by-step solution

Distribute exponent to both elements inside the parentheses

Using the rule of \((ab)^n = a^n * b^n\), \((-2 w)^{5}\) can be written as \((-2)^5 * (w)^5\).

Evaluate the expression

Evaluate \((-2)^5\) and \((w)^5\). \((-2)^5\) is -32 and \((w)^5\) is \(w^5\). Then replace them on the expression to get \( -32 * w^5\).

Key concepts

Exponents

When we deal with exponents in algebra, we're talking about the shorthand notation to express repeated multiplication of the same number, called the base, by itself. An expression like \(a^n\) reads as 'a' raised to the power of 'n', where 'a' is the base and 'n' is the exponent, indicating that 'a' is to be multiplied by itself 'n' number of times.

For example, \(2^3\) means 2 multiplied by itself 3 times, which equals 8. Negative bases raised to an odd exponent, such as \((-2)^5\), result in a negative product since an odd number of negative factors will always yield a negative result. Meanwhile, if that exponent were even, the result would be positive. This subtle difference is crucial when simplifying algebraic expressions with exponents.

Properties of Exponents

Understanding the properties of exponents is key to simplifying algebraic expressions efficiently. There are several rules we follow:

• The 'Product of Powers' rule: When multiplying two expressions with the same base, you can simply add the exponents, like \(a^m * a^n = a^{m+n}\).
• The 'Power of a Power' rule: When an exponent is raised to another exponent, such as \((a^m)^n\), you multiply the exponents together, resulting in \(a^{m*n}\).
• The 'Negative Exponent' rule states that \(a^{-n} = \frac{1}{a^n}\), which is the reciprocal of the base raised to the positive exponent.

The exercise given utilizes the 'Power of a Product' rule: \((ab)^n = a^n * b^n\). This property allows us to distribute the exponent over the multiplication inside the parentheses. Applying this property helps break down complex expressions into simpler, more manageable pieces, making them easier to evaluate.

Algebraic Operations

Algebra is full of operations that might seem daunting at first, but they are fundamentally extensions of basic arithmetic. Algebraic operations include addition, subtraction, multiplication, division, and exponentiation involving variables and constants.

When simplifying expressions like \((-2w)^5\), we perform algebraic operations step by step. First, we distribute the exponent to both the numerical coefficient and the variable separately, as separation is the crux of simplification. Once separated, we evaluate or simplify each part individually. After evaluating numerical parts like \((-2)^5\) to -32, we combine it back with the variable part, which has been similarly processed. Coefficient numbers and variables are treated like distinct entities that are operated on separately and then recombined to create a simplified expression.

It's essential to handle one operation at a time and follow the proper order often summarized by the acronym PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction). This orderly approach ensures accurate simplification of algebraic expressions and helps students avoid common mistakes.