Problem 72 - Elementary and Intermediate Algebra

Question

Use the power of a product rule for exponents to simplify each expression. $$ \left(-2 x^{2} y^{4}\right)^{5} $$

Step-by-Step Solution

Answer

The expression simplifies to \\(-32x^{10}y^{20}\\).

Step 1: Apply the Power of a Product Rule

The power of a product rule states that \( (ab)^n = a^n b^n \). This means that each factor within the parentheses will be raised to the power independently. For \( (-2x^2y^4)^5 \), this becomes \( (-2)^5 (x^2)^5 (y^4)^5 \).

Step 2: Simplify Each Part Separately

First, simplify \( (-2)^5 \): This is \( (-2) imes (-2) imes (-2) imes (-2) imes (-2) = -32 \). Next, simplify \( (x^2)^5 \): Using the power of a power rule \( (a^m)^n = a^{m imes n} \), we get \( x^{2 imes 5} = x^{10} \). Finally, simplify \( (y^4)^5 \): This becomes \( y^{4 imes 5} = y^{20} \).

Step 3: Combine Simplified Parts

Combine all the simplified parts from the previous step, which results in \(-32x^{10}y^{20}\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation involving the raising of a base number to a certain power or exponent. The operation is fundamental in algebra and is used to express repeated multiplication of the same number. For instance, if you see something like \( a^n \), that simply means you are multiplying \( a \) by itself \( n \) times.

In the expression \( (-2 x^2 y^4)^5 \), the entire term within the parentheses is treated as a single unit, and the exponent outside tells us to multiply it by itself five times. This process might seem complex when multiple variables and coefficients are involved, but using exponentiation rules helps simplify it greatly. Each component like \( -2 \), \( x^2 \), and \( y^4 \) can also be raised to the power of 5 according to established rules.

Algebraic Simplification

Simplifying algebraic expressions involves reducing them into their simplest form. This is achieved by applying various algebraic laws and rules, such as combining like terms and using exponent rules. Simplification makes an expression easier to understand and work within equations, solving and graphing.

When faced with \( (-2x^2y^4)^5 \), simplification has the main goal of expressing the expression without parentheses, whilst ensuring each element is independently adjusted according to the exponent. The process begins by breaking down the product inside the parentheses and raising each component separately to the power given. This step-by-step approach helps ensure that every part of the expression is accounted for and accurately incorporated.

Power Rule for Exponents

The Power Rule for Exponents is a crucial tool in simplifying expressions like \( (-2x^2y^4)^5 \). This rule says that when you have a power raised to another power, you multiply the exponents: \( (a^m)^n = a^{m \times n} \).

Applying this to our example, each component inside the parentheses \( -2 \), \( x^2 \), and \( y^4 \) is treated as separate entities, and we apply the power of 5 to each:

• For \( -2 \), you raise it to the 5th power: \( (-2)^5 = -32 \).
• For \( x^2 \), use \( (x^2)^5 \) resulting in \( x^{10} \).
• For \( y^4 \), use \( (y^4)^5 \) resulting in \( y^{20} \).

Once simplified, these components are multiplied back together without the need for repeated calculations, ultimately leading to a much simpler expression: \(-32x^{10}y^{20}\).

This systematic approach not only clarifies the expression's simpler structure but also ensures that calculations are efficiently carried out with precision.