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Chapter 10: Problem 49

Simplify the expression. $$\left(6 a^{2} b\right)^{2}$$

Short Answer

Expert verified
The simplified expression is \ 36 a^{4} b^{2} \.

Step by step solution

01

Understand the Expression

The given expression is \((6 a^{2} b)^{2}\). This is a power of a product, which means each factor inside the parentheses will be squared.
02

Apply the Power Rule

Use the power rule \((xy)^n = x^n y^n\) to distribute the exponent to each factor inside the parentheses: \[ ((6 a^{2} b)^{2}) = (6)^{2} (a^{2})^{2} (b)^{2} \]
03

Simplify Each Factor

Simplify each factor individually: \[(6)^{2} = 36, \, (a^{2})^{2} = a^{2 \times 2} = a^{4}, \, (b)^{2} = b^{2} \]
04

Combine the Simplified Factors

Combine the simplified factors to get the final expression: \[ 36 a^{4} b^{2} \]

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Key Concepts

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

power rule in algebra
When handling expressions with exponents, you will often use the power rule. This rule helps you simplify powers of a product. When you raise a product to a power, the power must be applied to each factor inside the parentheses. For example, in the expression \( (xy)^n \), you distribute the exponent $$n$$ both to $$x$$ and $$y$$, resulting in \( x^n y^n \).
Applying this rule is essential when simplifying more complex algebraic terms. Understanding this principle will make working with exponents much more manageable.
multiplying exponents
Multiplying exponents can be tricky but is straightforward with the correct approach. Consider the expression \( (a^m)^n \). To find the simplified form, multiply the exponents together. For instance, \( (a^2)^3 \) becomes \( a^{2 \times 3} = a^6 \).
It's crucial to remember this rule when simplifying expressions. Combining it with the power rule allows you to break down and reassemble algebraic expressions correctly and efficiently.
This principle is key when dealing with algebraic problems involving variables raised to various powers.
algebraic expressions in prealgebra
In prealgebra, you are introduced to simplifying algebraic expressions. This foundational skill involves combining like terms and using the distributive property. Simplifying helps make complex expressions easier to work with and understand.
Basic steps include:
  • Identifying like terms
  • Applying the distributive property
  • Combining terms and simplifying exponents

Developing these skills early ensures more complex tasks in higher-level algebra become less intimidating.

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Most popular questions from this chapter

Simplify. Write the answers with positive exponents only. $$\left(p^{4}\right)^{-5}$$

Subtract the polynomials. $$\left(-19 a^{5}-6 a^{3}+2 a^{2}+7\right)-\left(4 a^{5}+2 a^{3}+6 a^{2}-3\right)$$

The cost (in dollars) to rent a car is given by the polynomial \(29.99 x+0.40 y .\) In this context, \(x\) is the number of days that the car is rented and \(y\) is the number of miles driven. a. Evaluate the polynomial for \(x=3\) and \(y=350 .\) Interpret the answer in the context of this problem. b. Evaluate the polynomial for \(x=7\) and \(y=720 .\) Interpret the answer in the context of this problem.

Subtract the polynomials. $$\left(2 m n^{3}+6 m^{2} n^{2}+9 m n^{2}-3 m n\right)-\left(5 m n^{3}-2 m^{2} n^{2}-7 m n\right)$$

$$\text { Fill in the blanks. }\left(y^{2}\right)^{7}=y \square \cdot \square=y^{14}$$
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