Problem 41 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Then simplify the answer. $$ \frac{4 \cdot 2^{5}}{16-4^{2}+1} $$

Step-by-Step Solution

Answer

The simplified answer to the expression is 128.

Step 1: Execute the operation in the exponent

Here, \(2^{5}\) equals 32. So replace \(2^{5}\) with 32 in the numerator resulting in: \(4 \cdot 32\).

Step 2: Perform the multiplication in the numerator

Multiplying 4 and 32, you get 128.

Step 3: Resolve the exponentiation in the denominator

Solve \(4^{2}\) which equals 16. So the denominator now changes to: \(16 - 16 + 1\).

Step 4: Perform subtraction and addition in the denominator

Subtract and then add to simplify the denominator. The operation results in \(16 - 16 + 1 = 1\).

Step 5: Simplify the overall fraction

With 128 in the numerator and 1 in the denominator, the expression simplifies to: \(128/1\).

Step 6: Final simplification

Simplify the fraction. Since any number divided by 1 is the same number, the expression simplifies to 128.

Key Concepts

Understanding Exponents

Exponents are a key mathematical concept that signify the number of times a number, called the base, is multiplied by itself. When you see an expression like \(2^5\), the 2 is the base and 5 is the exponent. This tells us that 2 needs to be multiplied by itself 5 times: \(2 \times 2 \times 2 \times 2 \times 2\), which equals 32.

• The process of calculating an exponent is called exponentiation.
• It allows us to express large numbers in a more compact form.

In the given exercise, the exponent \(2^5\) was tackled first. Once simplified to 32, it streamlined the rest of the calculation process. Remember, working with exponents necessitates precision, especially in larger arithmetic expressions.

Working with Fractions in Expressions

Fractions are a way of representing numbers that aren't whole, involving a numerator and a denominator. In algebra, fractions often appear within complex expressions requiring simplification. In the expression \( \frac{4 \cdot 2^{5}}{16-4^{2}+1} \), we deal with a fraction that needs careful evaluation of both the numerator and denominator.

• The numerator is the upper part of the fraction, handled separately from the denominator, which is below the line.
• Once individual parts are simplified, they can be combined into a simple fraction to provide clear results.

The tricky part about fractions within expressions is ensuring the order of operations is respected. For example, exponentiation should be performed before multiplication in the numerator, and all components of the denominator should be individually resolved before division takes place. This attention to detail simplifies the process and ensures you arrive at the correct answer.

Simplifying Complex Expressions

Simplifying expressions involves breaking down a complex mathematical phrase into its simplest form. This process is key to solving equations efficiently. In our exercise, the expression \( \frac{128}{1} \) was the end result of simplifying the given complex expression.

• Firstly, simplify any operations within exponents to manage large numbers.
• Follow the order of operations, PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) to accurately simplify step-by-step.
• Combine like terms and reduce fractions where possible.

In the final step, the fraction \( \frac{128}{1} \) was simplified. Since any number divided by 1 is the number itself, the expression simplified to 128. Simplification helps not only to find the right answer but also makes understanding the solution clearer. It encapsulates the essence of breaking down complex problems into approachable steps.