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Chapter 4: Problem 38

Simplify. $$-6\left(p^{2}\right)^{4}+10 p^{8}$$

Short Answer

Expert verified
The simplified form of \( -6\left(p^{2}\right)^{4}+10 p^{8} \) is \( 4p^{8} \).

Step by step solution

01

Simplify the first term using rules of exponents

Use the rule \( (a^{m})^{n} = a^{mn} \) power of a power rule for exponents to simplify \( -6\left(p^{2}\right)^{4} \) into \( -6p^{2*4} = -6p^{8} \).
02

Combine the terms

Now we combine \( -6p^{8} + 10p^{8} \), which have the same base and exponent, using the additive rule of polynomials to get \( 4p^{8} \).

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Key Concepts

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

Understanding Exponents
Exponents are a fundamental part of algebra and serve as a shorthand way of expressing repeated multiplication. For instance, rather than writing \( p \times p \), we write \( p^2 \). Here, "2" is the exponent, indicating that the base, \( p \), is multiplied by itself. Exponents are key in representing powers and can significantly simplify expressions.
  • **Base**: The number or variable that is being multiplied by itself.
  • **Exponent**: Shows how many times the base is used as a factor.
When dealing with exponents, understanding the base and the exponent is crucial for manipulating algebraic expressions effectively.
The Power of a Power Rule
The power of a power rule is a useful property in algebra that helps simplify expressions involving exponents. It states that when you raise an exponent to another exponent, you multiply the exponents together. The rule can be written as \((a^m)^n = a^{m \cdot n}\).
  • For example: \((p^2)^4\) simplifies to \(p^{2 \cdot 4} = p^8\).
This rule is extremely handy when simplifying polynomials, as it reduces complex expressions into simpler ones by converting multiple layers of exponentiation into a single expression.
The Additive Rule for Polynomials
The additive rule for polynomials allows us to combine like terms, which means terms that have the same base and exponent. When terms are like, they can be added or subtracted by simply combining their coefficients, while the variables and their exponents remain unchanged. Consider the expression:
  • \(-6p^8 + 10p^8\)
Using the additive rule, these can be combined as follows: \(-6 + 10 = 4\), hence \(-6p^8 + 10p^8 = 4p^8\). The coefficients, \(-6\) and \(10\), are the only parts that change when adding or subtracting, simplifying the polynomial further.
Simplifying Polynomials with Steps
Simplifying polynomials involves reducing them to their simplest form by combining like terms and applying rules of exponents. It's a process that makes calculations and interpretations more manageable. Here's a step-by-step guide:
  • Use the power of a power rule to simplify terms within parentheses: \(-6(p^2)^4\) becomes \(-6p^8\).
  • Identify like terms, which have the same base and exponent.
  • Combine these like terms using the additive rule: \(-6p^8 + 10p^8\) becomes \(4p^8\).
The goal of simplifying is to make the expression easier to work with while preserving its original value. A well-simplified polynomial is concise and efficient for calculation and analysis.

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Most popular questions from this chapter

Perform the operation. Subtract \((2 x+5 y)\) from \((5 x-8 y)\)

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