Problem 32 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ \left[(-5 x y)^{2}\right]^{5} $$

Step-by-Step Solution

Answer

\[(-5)^{10}\cdot x^{10}\cdot y^{10}\]

Step 1: Apply the Power of a Power Rule

The power of a power rule states that \((a^m)^n = a^{m \cdot n}\). Apply this rule to \(\left[(-5 x y)^{2}\right]^{5}\) to get \((-5xy)^{2 \cdot 5}\).

Step 2: Simplify the Exponent

The total power becomes \(2 \cdot 5 = 10\), so the equation now is \((-5xy)^{10}\).

Step 3: Use the distributive property

Now apply the distributive rule which states that \((ab)^n = a^n \cdot b^n\). The equation \((-5xy)^{10}\) becomes \((-5)^{10}(x)^{10}(y)^{10}\).

Key Concepts

Understanding Exponents

When we talk about exponents, we're referring to the shorthand way of expressing repeated multiplication of the same factor. In an expression like \(a^m\), \(a\) represents the base and \(m\) is the exponent, which tells us how many times to multiply \(a\) by itself.
To understand the problem \(\left[(-5 x y)^{2}\right]^{5}\), imagine having \((-5xy)\) multiplied together twice and then taking that result and multiplying it by itself five more times. Sounds exhausting, right? That's where the power of a power rule simplifies the process, reducing it to a single exponentiation step.

Applying the Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions where a single term is being multiplied by a sum (or difference) of terms. For exponents, it helps us distribute a common exponent to each base when multiple bases are multiplied together under one exponent, like in \((ab)^n = a^n \cdot b^n\).
When applied to our problem, \((-5xy)^{10}\), we distribute the exponent of 10 to each component of the base: \((-5)\), \(x\), and \(y\), resulting in \((-5)^{10}\), \(x^{10}\), and \(y^{10}\). This breaks down a complex expression into a simpler, more manageable form.

Simplifying Expressions

To simplify an expression means to make it as straightforward as possible. This often involves reducing the number of terms, combining like terms, or applying algebraic rules to present an equivalent expression in a more concise manner. Simplifying can involve various operations including the use of exponents and the distributive property as we saw in the exercise.
With \((-5xy)^{10}\), we are not just multiplying numbers, but also keeping in mind the variables involved. Each part of this expression can be simplified separately, taking a seemingly complex algebraic expression and making it understandable by focusing on one part at a time.

Algebraic Rules in Practice

Algebraic rules are the toolkit for manipulating expressions and equations. They include the power of a power rule, which tells us how to handle an expression raised to an exponent when the expression itself contains an exponent. These rules are systematic and predictable, allowing us to simplify expressions and solve equations with confidence.
In our exercise, after applying the power of a power rule and the distributive property, we arrive at \((-5)^{10}(x)^{10}(y)^{10}\). This showcases how algebraic rules help in systematically breaking down and simplifying complex expressions.