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Chapter 10: Problem 3

Perform the indicated operations. $$\left(2 x^{2} y\right)^{4}$$

Short Answer

Expert verified
16x^{8}y^{4}

Step by step solution

01

- Understand the problem

The given expression is \( (2 x^{2} y)^{4} \). We need to simplify this expression by performing the exponentiation.
02

- Apply the power rule

According to the power rule, \( (a \cdot b)^{n} = a^{n} \cdot b^{n} \). So we can split the expression into individual powers: \( (2)^{4} \cdot (x^{2})^{4} \cdot (y)^{4} \).
03

- Simplify each term

Calculate each term separately: \( (2)^{4} = 2 \times 2 \times 2 \times 2 = 16 \, (x^{2})^{4} = x^{2 \cdot 4} = x^{8} \, (y)^{4} = y^{4} \).
04

- Combine simplified terms

Combine the simplified terms: \( 16 \cdot x^{8} \cdot y^{4} = 16x^{8}y^{4} \).

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Key Concepts

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

power rule in algebra
The power rule in algebra is essential for simplifying expressions with exponents. The rule states: * When you have a product raised to a power, apply the exponent to each factor inside the parentheses separately.
* This means, for an expression like \((ab)^n\), you can distribute the exponent n to both a and b.
* Mathematically, this is represented as \((ab)^n = a^n \. b^n\).
In our exercise, the given expression is \((2x^2 y)^4\). We use the power rule to split it into simpler parts: \((2)^4 \. (x^2)^4 \. (y)^4\).
We can see how this rule helps in breaking down complex expressions into manageable pieces.
simplifying expressions
Simplifying expressions involves reducing them to their simplest form. This is often done by performing operations like multiplication and applying rules such as the power rule. In the context of the given exercise: To simplify \((2x^2 y)^4\), we apply the power rule first.
Then, we calculate the power of each term separately:
\((2)^4 = 2 \. 2 \. 2 \. 2 = 16\)

For the variable x: \((x^2)^4 = x^{2\cdot4} = x^8\)

Finally, for the variable y: \((y)^4 = y^4\).
Always follow these steps to simplify each part before combining them.
algebraic operations
Algebraic operations are the building blocks of simplifying expressions. They include addition, subtraction, multiplication, division, and exponentiation. In our exercise, we focused on multiplication and exponentiation. Here鈥檚 a quick overview:
* **Multiplication:** Combining numbers or variables by multiplying them. Example: \(2 \cdot 3 = 6\).
* **Exponentiation:** Raising numbers or variables to a power. Example: \((x^2)^3 = x^6\).
In the step-by-step solution, we saw that \(2^4 = 16\), \((x^2)^4 = x^8\), and \((y)^4 = y^4\).
These operations are then combined to give the final result: \16x^8 y^4\.
Understanding these operations is crucial for mastering algebra.

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Most popular questions from this chapter

Perform the indicated operations. $$\left(-5 a^{2}-9 a^{3}+10 a^{4}\right)+\left(2 a^{3}-a^{4}\right)-\left(8 a+4 a^{3}+10 a^{4}\right)$$

Evaluate the polynomial for the given values of the variable. \(3 c^{3}-c+8\) a. for \(c=-1\) b. for \(c=0.1\)

Perform the indicated operations. $$\left(-\frac{5}{4} t^{3}+\frac{1}{8} t+1\right)-\left(\frac{3}{2} t^{3}-\frac{1}{2} t+\frac{2}{3}\right)$$

Multiply the polynomials. $$\begin{array}{r}9 a^{2}+2 a-4 \\\\\times \quad 4 a^{2}+a+3 \\\\\hline\end{array}$$

Simplify the expression. $$\left(\frac{2 w}{p}\right)^{4}$$
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