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Chapter 9: Problem 62

Simplify. $$\left(2 x^{3} y^{4}\right)^{5}$$

Short Answer

Expert verified
The simplified form of \( (2x^{3}y^{4})^5 \) is \(32x^{15}y^{20}\)

Step by step solution

01

Apply the Power of a Power Rule

To simplify the expression \( (2x^{3}y^{4})^5 \), apply the power of a power rule \( (a^m)^n = a^{m*n} \) to each part of the expression. For the number 2, apply the power 5, giving \(2^5\). For the variables \(x\) and \(y\), the existing exponents (3 and 4 respectively) are multiplied by 5, giving \(x^{3*5}\) and \(y^{4*5}\).
02

Simplify the Exponents

Now simplify the expressions. \(2^5\) equals 32. \(x^{3*5}\) equals \(x^{15}\), and \(y^{4*5}\) equals \(y^{20}\)
03

Combine the Simplified Parts

Combine the simplified parts to get the final simplified expression: \(32x^{15}y^{20}\)

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Key Concepts

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

Power of a Power Rule
The Power of a Power Rule is a fundamental concept in algebra that helps simplify expressions involving exponents. When you have a power raised to another power, such as \((a^m)^n\), the rule tells us to multiply the exponents: \(a^{m\cdot n}\). This simplifies the power expression more easily.
Let's take an example. Suppose you have the expression \((x^3)^4\). By using the Power of a Power Rule, multiply the exponents 3 and 4, resulting in \(x^{12}\).
  • This rule helps break down complex power expressions into simpler parts.
  • Always look for opportunities to apply this rule when you see nested exponents.
Simplifying Expressions
Simplifying expressions is all about breaking down complex algebraic expressions into simpler and more manageable forms. It's like solving a puzzle. You apply rules and properties step by step to make the expression easier to understand and work with.
When simplifying an expression such as \((2x^{3}y^{4})^5\), there's a systematic approach:
  • First, identify all parts of the expression affected by exponents.
  • Apply any pertinent exponent rules, like the Power of a Power Rule, to reduce each component.
  • Perform arithmetic operations like multiplication and division where necessary.
For example, simplify each part: calculate \(2^5 = 32\), \(x^{3\cdot5} = x^{15}\), and \(y^{4\cdot5} = y^{20}\). Finally, combine these simplified components to express your final answer.
Algebraic Expressions
Algebraic expressions consist of numbers, variables, and operators (like +, -, *, /) structured in a meaningful way. Understanding how to manipulate them is crucial as they form the foundation of algebra.
Algebra lets you express complex calculations and relationships in a compact form, allowing for generalization and use of mathematical logic. An algebraic expression can range in complexity from a simple \(2x\) to a more involved \((2x^{3}y^{4})^5\).
  • Variables represent unknowns and allow us to formulate equations to solve problems.
  • Operators determine how the variables and numbers interact in the expression.
  • Expressions can be simplified, factored, expanded, and solved depending on their form and the problem context.
Remember that when working with algebraic expressions, clear understanding of the components and rules鈥攍ike those related to exponents鈥攅nables you to approach problems with confidence.

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