Problem 17 - Algebra 1

Question

Use the power of a product property to simplify the expression. $$ \left(a b^{2}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified form of the expression \((ab^{2})^{2}\) is \(a^2 b^4\)

Step 1: Recognize the structure

In the given expression \((a b^{2})^{2}\), 'a' and 'b^2' are multiplied and the result is then squared. This fits to the power of a product rule.

Step 2: Apply the power of a product rule

According to the rule, the given equation becomes \(a^2 \cdot (b^2)^2\).

Step 3: Final simplification

Taking square of both 'a' and 'b^2' we get, \(a^2 \cdot b^{2*2}\) which simplifies to \(a^2 \cdot b^4\).

Key Concepts

Power of a Product

The power of a product property is a fundamental rule in algebra that helps us manage exponents more effectively. This property is particularly useful when dealing with expressions that involve products raised to an exponent. To break it down simply:

• When you have an expression in the form of \((xy)^n\), this is equivalent to raising each factor inside the parentheses to the power separately. In other words, \((xy)^n = x^n \cdot y^n\).

This might initially seem a little complex, but it becomes second-nature with practice. For example, if you are given \((a b^2)^2\), according to the power of a product property, you can distribute the exponent of 2 to both elements inside the parentheses, leading to \(a^2 \cdot (b^2)^2\). This simplifies the computation and paves the way for further simplification.

Exponents

Exponents are a way to simplify repeated multiplication. When you see something like \(a^n\), it means that 'a' is multiplied by itself 'n' times. Understanding the rules of exponents can drastically simplify calculations that might otherwise become cumbersome.

• The key rules of exponents include the power rule, which we used in our example. When you raise a power to another power, you multiply the exponents: \((b^m)^n = b^{m \cdot n}\).
• Additionally, knowing that any number to the power of zero is one (except zero itself) and that raising a number to the first power yields the number itself, holds significant value in simplifying expressions.

In our specific exercise, \((b^2)^2\) applies the power rule to simplify into \(b^{2 \cdot 2} = b^4\). Hence, understanding these fundamental rules can make exponent calculations fast and straightforward.

Expression Simplification

Expression simplification aims to reduce equations to their simplest form while maintaining their equality. This process often involves familiar operations such as distributing exponents using properties like the power of a product, simplifying powers, and addressing coefficients.

• The main focus is to transform the given expression into its simple form by applying algebraic rules efficiently.
• In our example, starting from \((a b^2)^2\), we first used the power of a product property to separately raise 'a' and 'b^2' to the second power. This lead to \(a^2 \cdot b^{4}\).

This kind of simplification is crucial in mathematical problem solving because it allows us to see relationships between numbers more clearly and makes further operations or equations a lot easier to manage. Mastery of this concept is essential for progressing in more advanced areas of math, as it provides a basis for solving increasingly complex problems with confidence and efficiency.