Problem 42 - Algebra 1

Question

Write your answer as a power or as a product of powers. $$ (-a b)\left(a^{2} b\right)^{2} $$

Step-by-Step Solution

Answer

The simplified form of expression \(-a b(a^{2} b)^{2}\) is \(-a^{5}b^{3}

Step 1: Expand the Expression

First, take the expression \(-a b(a^{2} b)^{2}\) and apply property of exponents which states that \((a^m)^n = a^{m*n}\). So, we get: \(-a b(a^{4}b^{2})\).

Step 2: Simplify the Expression

Next, apply the property of exponents which states that \( a^m * a^n = a^{m+n}\). So, \(-a b*a^{4}b^{2} = -a^{5}b^{3}\). Thus, the simplified form of given expression is \(-a^{5}b^{3}\).

Key Concepts

Properties of Exponents

Exponents are an elegant way to express repeated multiplication. Understanding their properties allows us to simplify complex expressions easily.
One fundamental property is the power of a power rule:

• When you raise a power to another power, you multiply the exponents. In symbols, \((a^m)^n = a^{m\times n}\).

This is used when you have expressions like \((a^2b)^2\), allowing you to rewrite it as \(a^{2\times 2}b^2 = a^4b^2\).
Another key property is the product of powers property, which states:

• When multiplying two expressions with the same base, you add their exponents.In other words, \(a^m \times a^n = a^{m+n}\).

This property helps combine terms like \(a \times a^4\), resulting in \(a^{1+4} = a^5\).
These properties make handling exponents swift and straightforward once mastered. Apply them frequently to gain the confidence needed for algebraic processes.

Simplifying Expressions

Simplifying expressions in algebra involves reducing them to their simplest form. This process often includes applying the properties of exponents.
The goal is to make expressions manageable and easier to understand.
Let's consider the expression \(-a b(a^{4}b^{2})\). Upon utilizing the properties of exponents, the expression transforms step-by-step into \(-a^{5}b^{3}\).
When simplifying, remember:

• Identify like terms (same base and same variables).
• Combine them using exponent rules to sum up powers or reduce them as needed.

Careful attention to these rules results in tidy expressions that are easy to interpret.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They serve as the backbone for many algebraic problems.
Expressions like \((-a b)(a^{2} b)^{2}\) involve several operations and require strategic approaches to solve.

• First, understand each component of the expression: variables (like \(a\) and \(b\)), constants, and the operations (multiplication and exponentiation).
• Use known algebraic rules, such as the properties of exponents, to manipulate and simplify the expression.

Practicing with different expressions helps grasp the versatility of algebra and builds problem-solving skills.
The more proficient you become with expressions, the easier algebra will be.