Problem 201 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{2^{3}+4^{2}}{\left(\frac{2}{3}\right)^{2}} $$

Step-by-Step Solution

Answer

54

Step 1 - Simplify the numerator

First, simplify the expression in the numerator: \(2^3 + 4^2\). \(2^3 = 2 \times 2 \times 2 = 8\). \(4^2 = 4 \times 4 = 16\). So, the numerator becomes \(8 + 16 = 24\).

Step 2 - Simplify the denominator

Now, simplify the denominator: \(\left(\frac{2}{3}\right)^2\). To square a fraction, square both the numerator and the denominator: \(\left(\frac{2}{3}\right)^2 = \frac{2^2}{3^2} = \frac{4}{9}\).

Step 3 - Divide the simplified numerator by the simplified denominator

Next, divide the numerator by the denominator: \(\frac{24}{\frac{4}{9}}\). This is the same as multiplying by the reciprocal of the denominator: \(24 \times \frac{9}{4}\).

Step 4 - Multiply and simplify

Finally, compute the multiplication: \(24 \times \frac{9}{4}\). First, multiply the numerators: \(24 \times 9 = 216\). Then, divide by the denominator: \(\frac{216}{4} = 54\).

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of a number by itself. For example, in the step-by-step solution, we see expressions like \(2^3\) and \(4^2\). The exponent tells us how many times to multiply the base number by itself.

• \(2^3\) means \(2 \times 2 \times 2 = 8\).
• \(4^2\) means \(4 \times 4 = 16\).

These skills are crucial for simplifying algebraic expressions that contain powers and exponents. Always handle exponents first before moving on to addition or subtraction.

Working with Fractions

Fractions represent parts of a whole. In the problem, we simplified the denominator by squaring the fraction \(\left(\frac{2}{3}\right)^2\).
When squaring a fraction:

• Square the numerator: \(2^2 = 4\).
• Square the denominator: \(3^2 = 9\).

Therefore, \(\left(\frac{2}{3}\right)^2 = \frac{4}{9}\). Understanding how to manipulate fractions is key to solving more complex problems, especially when they are part of larger expressions.

Integer Operations

Integer operations involve basic arithmetic functions, such as addition, subtraction, multiplication, and division, with whole numbers. In the solution, the integer operations are:

• Addition: In the numerator, \(8 + 16 = 24\).
• Multiplication: \(24 \times 9 = 216\).
• Division: \(\frac{216}{4} = 54\).

These steps demonstrate how arithmetic rules apply when dealing with integers in algebraic expressions. Each operation must be performed in the correct order to simplify the expression properly.

Understanding Reciprocals

A reciprocal is simply the inverse of a number or fraction. It's what you multiply by to get 1. For example, the reciprocal of \(\frac{2}{3}\) is \(\frac{3}{2}\).
In the solution, the reciprocal comes into play when dividing fractions:

• To divide by a fraction, multiply by its reciprocal.
• For \(\frac{24}{\frac{4}{9}}\), multiply by the reciprocal of \(\frac{4}{9}\), which is \(\frac{9}{4}\).

Therefore, \(24 \times \frac{9}{4}\) simplifies to \(\frac{216}{4} = 54\). Understanding reciprocals helps greatly in performing operations involving divisions of fractions.