Problem 29 Simplify each of the following a... [FREE SOLUTION]

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Understanding Elementary Algebra with Geometry

Lewis Hirsch, Arthur Goodman

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 7: Problem 29

Simplify each of the following as completely as possible. $$\left(x^{2} y\right)^{2}$$

Short Answer

Expert verified

$x^4 y^2$

Step by step solution

Apply the Power Rule

The Power Rule states that $(a^m)^n = a^{mn}$. Use this rule to simplify $(x^2 y)^2$.

Distribute the Exponent

Distribute the exponent of 2 to each factor inside the parentheses: $(x^2)^2 (y)^2$.

Simplify Each Term

Using the Power Rule again, simplify each term individually: $(x^2)^2 = x^{2 \times 2} = x^4$ and $y^2 = y^2$. Combine to get \x^4 y^2\.

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Key Concepts

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

Power Rule

The Power Rule is an essential tool in algebra that helps simplify expressions with exponents. According to this rule, when you raise a power to another power, you multiply the exponents. This can be written as $(a^m)^n = a^{mn}$.

Let's use this rule with our example: We start with $(x^2 y)^2$. Using the Power Rule, we can distribute the exponent over each term inside the parentheses. This turns into $(x^2)^2 (y)^2$.

Notice how the Power Rule simplifies expressions with nested exponents easily. It's like unwrapping layers by multiplying the exponents together. This makes it a handy tool in algebraic simplification.

Exponent Distribution

Exponent distribution is the process of applying an exponent to each term inside a parenthesis. It's a key step in simplifying complex algebraic expressions.

In our example, we started with $(x^2 y)^2$. To distribute the exponent of 2, we applied it to each factor inside the parentheses: $x^2$ and $y$. This gave us $(x^2)^2 (y)^2$.

By doing this, we made it easier to manage each term individually. For instance, our expression becomes a series of simpler problems where each term is raised to a particular power. This is a straightforward technique to break down complicated expressions.

Mathematical Simplification

Mathematical simplification involves reducing an expression to its simplest form. This usually involves combining like terms, using arithmetic rules, and simplifying powers and exponents.

Returning to our example, we have $(x^2)^2 (y)^2$. To simplify, we use the Power Rule again on $(x^2)^2$, which translates to $x^{2 \times 2} = x^4$. Meanwhile, $y^2$ remains the same because it is already simplified. Putting it all together, our final simplified expression is $x^4 y^2$.

Remember, simplification helps make expressions easier to work with. By consistently applying rules like the Power Rule and proper exponent distribution, complex expressions become manageable.

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91影视

Simplify each of the following as completely as possible. $$\left(x^{2} y\right)^{2}$$

Short Answer

Step by step solution

Apply the Power Rule

Distribute the Exponent

Simplify Each Term

Key Concepts

Power Rule

Exponent Distribution

Mathematical Simplification

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