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Magazine Chapter 5: Problem 10
Simplify: \(\left(-3 x y^{4}\right)^{5}\)
Short Answer
Expert verified
The simplified form is
\(-243x^5y^{20}\).
Step by step solution
01
Apply the Power of a Product Rule
The expression inside the parentheses is \(-3x y^4\). When an expression in the form \((ab)^n\) is raised to a power, the power \(n\) applies to all components inside the parentheses. For our expression, we distribute the power of 5 to each element: \((-3)^5\), \(x^5\), and \((y^4)^5\).
02
Simplify the Constants
First, calculate \((-3)^5\). The result is \(-3 imes -3 imes -3 imes -3 imes -3 = -243\). So, \((-3)^5 = -243\).
03
Apply the Power Rule to the Variables
For the variable \(x\), since we have \(x^5\), it remains \(x^5\). For \(y^4\), apply the power rule of exponents: \((y^m)^n = y^{m imes n}\). Here, it is \((y^4)^5 = y^{4 imes 5} = y^{20}\).
04
Combine the Simplified Parts
Putting it all together, the simplified expression from the previous steps is \(-243x^5y^{20}\). This is the final simplified form of the original expression.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power of a Product Rule
The Power of a Product Rule is a helpful tool when working with expressions that have powers applied to products. This rule states that, when you have a product inside a set of parentheses raised to an exponent like \[(ab)^n\], the exponent applies to each element of the product separately. For example:
- If you have \((ab)^n\), it becomes \((a^n)(b^n)\).
- This means that each individual factor within the parentheses is raised to the power specified outside the parentheses.
Simplifying Expressions
Simplifying expressions is about breaking down a complex expression into its most basic form. When simplifying, it's important to consider all components of the expression. This usually involves:
- Applying algebraic rules and laws, like the Power of a Product Rule or the Power Rule of Exponents, to enable easier calculations.
- Performing arithmetic calculations on constants, like computing \(-3^5\) which results in \-243\.
- Reducing expressions to cleaner, more compact forms while ensuring correctness.
Exponents
Exponents are a shorthand way to express repeated multiplication of a number by itself. For instance, \(x^n\) means that the base, \(x\), is used as a factor \(n\) times. Understanding how to manipulate exponents is crucial in many levels of mathematics:
- Exponents help simplify expressions and equations by reducing lengthy multiplication into a simple term.
- They are guided by specific rules and properties, making calculations easier once these are understood.
- Negative bases with an odd exponent result in a negative product, as seen in \(-3^5\), which equals \-243\.
Power Rule of Exponents
The Power Rule of Exponents is key when raising a power to another power. If you have \( (a^m)^n \), the Power Rule tells you to multiply the exponents: \(a^{m \times n}\). This is vital for simplifying expressions involving exponents.In our example with \(y^4\), using the Power Rule, \( (y^4)^5\) becomes \( y^{4 \times 5} \) or \ y^{20}\. By applying this rule, what seems complicated can become quite simple:
- Identify where the Power Rule applies in your problem.
- Multiply the exponents for all terms affected, ensuring accuracy in calculations.
- Simplify each part and bring them together for the final expression.
