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Chapter 10: Problem 51

Simplify the expression. $$\left(-2 t v^{3}\right)^{4}$$

Short Answer

Expert verified
16t^4v^{12}

Step by step solution

01

Understand the power rule

When raising a product to a power, apply the power to each factor inside the parentheses. This means \( (ab)^n = a^n \times b^n \).
02

Apply the power to each term inside the parentheses

Apply the power of 4 to each term inside the parentheses: \( (-2tv^3)^4 = (-2)^4 \times (t)^4 \times (v^3)^4 \).
03

Simplify each term

Calculate each term separately: \( (-2)^4 = 16 \, \ t^4 = t^4 \, and \ (v^3)^4 = v^{12} \).
04

Write the final expression

Combine all the simplified terms to get the final expression: \( 16t^4v^{12} \).

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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 part of exponentiation. It allows you to simplify expressions where a product is raised to a power. The rule is simple: (ab)^n = a^n 脳 b^n This means you apply the exponent to each term inside the parentheses. This rule helps in breaking down complex expressions into more manageable parts. In our example, we started with (-2tv^3)^4. By applying the Power Rule: (-2)^4 脳 (t)^4 脳 (v^3)^4, we simplified the expression correctly. The Power Rule can be a useful tool in many algebraic operations, not just exponentiation, so mastering it is key.
Simplifying Expressions
Simplifying expressions is a core skill in algebra. It means breaking down expressions into their simplest form. You often need to apply rules like the Power Rule. In our example, we had (-2tv^3)^4. Apply the power of 4 to each term inside the parentheses. This turned it into (-2)^4 脳 (t)^4 脳 (v^3)^4. By simplifying each term step-by-step, we make the expression easier to work with. This method helps avoid mistakes and ensures that each part of the expression is correctly simplified.
Algebraic Operations
Algebraic operations include addition, subtraction, multiplication, and division, but they also extend to working with exponents and other more complex functions. Understanding how to perform these operations is crucial for solving algebraic expressions and equations. In our example, we used the Power Rule and then proceeded to multiply the simplified terms. First, calculate (-2)^4, which is 16. Next, raise 't' to the fourth power, giving us t^4. Lastly, raise (v^3) to the power of 4, giving us v^12. Put together, we get 16*t^4*v^12. Knowing how to properly conduct these operations ensures accurate results and understanding of algebraic principles.

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Most popular questions from this chapter

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Write a binomial with degree 4.

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