Problem 59 - Algebra 1

Question

Then evaluate the expression when \(a=1\) and \(b=2\). $$ \left(a \cdot b^{2}\right)^{2} $$

Step-by-Step Solution

Answer

The final result of evaluating the expression \((a \cdot b^{2})^{2}\) using \(a=1\) and \(b=2\) is 16

Step 1: Substitute the Given Values in the Expression

Replace the variable \(a\) with 1 and the variable \(b\) with 2 in the given expression to obtain \((1 \cdot 2^{2})^{2}\)

Step 2: Simplify Inside the Brackets

Calculate the value of \(2^{2}\) which equals 4, then multiply by \(a=1\) to obtain \((1 \cdot 4)^{2} = 4^{2}\)

Step 3: Calculate the Power

Calculate the value of \(4^{2}\) which equals 16

Key Concepts

Substitution

Substitution is a fundamental technique in algebra. It allows us to replace variables in algebraic expressions with their given values. This makes expressions easier to work with and understand. In the provided exercise, you are given the expression \((a \cdot b^{2})^{2}\), and told to evaluate it when \(a=1\) and \(b=2\). The first step in substitution is to clearly identify each variable in the expression and decide which values will replace them. Substitution transforms the original expression into a numerical form.

• Replace \(a\) with 1
• Replace \(b\) with 2

Applying these changes to the expression \((a \cdot b^{2})^{2}\) makes it \((1 \cdot 2^{2})^{2}\).This new numerical expression is now ready for further simplification and calculation. Always ensure substitution is done correctly; otherwise, all subsequent calculations might be incorrect.

Exponentiation

Exponentiation is the operation of raising a number to a power and is represented by the exponent notation such as \(b^{n}\). Raising a number to an exponent means multiplying it by itself a certain number of times specified by the exponent. In our exercise, exponentiation appears twice. Initially, we find \(b^{2}\), which involves multiplying the base \(b\) by itself. Given \(b=2\), this is calculated as:\[2^{2} = 2 \times 2 = 4\]The expression changes to \((1 \cdot 4)^{2}\) after successfully computing \(b^{2}\). The final step of exponentiation is performed on \(4\), raising it again to the power of \(2\):\[4^{2} = 4 \times 4 = 16\]Understanding exponentiation is crucial as it frequently appears in various math problems. It requires attention to the base and the power, ensuring calculations are done in the correct order.

Simplification

Simplification is all about making an algebraic expression easier to manage. It often involves performing calculations within parentheses, handling power operations, and reducing expressions to their simplest form. Simplification makes expressions clearer and their values easier to evaluate.In the given problem, the first simplification involves evaluating the expression within the brackets:

• Calculate power \(b^{2} = 4\) and then multiply by \(a=1\)

This gave us \((1 \cdot 4)^{2}\). The next simplification step is calculating the square of \(4\).By simplifying step-by-step:

• First simplify \(1 \cdot 4\) to get \(4\)
• Then simplify \(4^{2} = 16\)

Each step of simplification helps break the problem into manageable parts. It's important to follow the correct sequence, especially with operations involving parentheses and exponents, to reach the right solution.