Problem 76 - Brief Applied Calculus

Question

Simplify. $$ \frac{\left(5 x^{2} y^{3} z\right)^{2}}{5(x y z)^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \(5x^2y^4\).

Step 1: Simplify the Numerator

First, simplify the numerator \((5x^2y^3z)^2\). Apply the exponent to each term inside the parentheses. This gives us:\[(5x^2y^3z)^2 = 5^2 imes (x^2)^2 imes (y^3)^2 imes z^2 = 25x^4y^6z^2\]Thus, the numerator becomes \(25x^4y^6z^2\).

Step 2: Simplify the Denominator

Now, simplify the denominator \(5(xyz)^2\). Apply the exponent to the product inside the parentheses:\[5(xyz)^2 = 5 imes x^2 imes y^2 imes z^2\]Thus, the denominator becomes \(5x^2y^2z^2\).

Step 3: Divide and Simplify the Expression

Divide the simplified numerator by the simplified denominator:\[\frac{25x^4y^6z^2}{5x^2y^2z^2}\]Start by dividing the coefficients: \(\frac{25}{5} = 5\).Next, simplify the exponents:- For \(x\): \(x^{4-2} = x^2\)- For \(y\): \(y^{6-2} = y^4\)- For \(z\): \(z^{2-2} = z^0 = 1\)This leaves us with \(5x^2y^4\) after simplification.

Key Concepts

Exponents

Exponents, often called powers, are used to express repeated multiplication of the same number. For instance, in the expression \(a^b\), \(a\) is the base and \(b\) is the exponent. This tells us to multiply \(a\) by itself \(b\) times. Thus, \(3^2\) means \(3 \times 3 = 9\).

When dealing with expressions like \((5x^2y^3z)^2\), each element inside the parentheses is raised to the power of 2. This means you should apply the exponent to each term separately:

• For the number: \(5^2 = 25\).
• For the variables: \((x^2)^2 = x^{4}\), \((y^3)^2 = y^{6}\), and \(z^2\).

You end up with \(25x^4y^6z^2\). By systematically applying the exponent to each component, you're able to simplify complex expressions greatly.
Remember, exponents follow specific mathematical rules, such as:

• When multiplying similar bases, add the exponents: \(x^a \times x^b = x^{a+b}\).
• When raising an exponent to another power, multiply the exponents: \((x^a)^b = x^{a \times b}\).

Numerator and Denominator

In fractions, the term above the line is called the numerator, and the one below is known as the denominator.Understanding these parts is crucial because it helps in the operations such as simplifying fractions.

In the expression \(\frac{(5x^2y^3z)^2}{5(xyz)^2}\), the numerator is \((5x^2y^3z)^2\) and the denominator is \(5(xyz)^2\). Both need to be simplified separately first:

• For the numerator \((5x^2y^3z)^2 = 25x^4y^6z^2\).
• For the denominator \(5(xyz)^2 = 5x^2y^2z^2\).

By breaking down the components of fractions into numerators and denominators, you can simplify the fraction step by step.

Knowing how to manipulate the numerator and the denominator individually allows you to tackle more complicated algebraic operations with confidence.

Simplifying Fractions

Simplifying fractions is the process where you reduce the fraction to its simplest form. This helps to make expressions easier to work with, especially in more complex algebraic equations. Let's see how simplification works in our exercise.

You have a fraction: \[\frac{25x^4y^6z^2}{5x^2y^2z^2}\].The goal is to reduce it as much as possible by canceling common factors in the numerator and the denominator.

• First, divide the coefficients: \(\frac{25}{5} = 5\).
• Next, work with the variables by subtracting the exponents for each base:
  - For the base \(x\), subtract the exponent in the denominator from that in the numerator: \(x^{4-2} = x^2\).
  - For the base \(y\): \(y^{6-2} = y^4\).
  - For the base \(z\): \(z^{2-2} = z^0\), which simplifies to 1 (since any number to the power of 0 is 1).

The simplified form is \(5x^2y^4\).
Understanding and practicing simplification can greatly aid in demystifying complex mathematical expressions. It allows for clearer insight into the behavior of equations, making them easier to solve or manipulate for further calculations.