Problem 51 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \left(\frac{x^{-2} y^{4} z^{0}}{x^{3} y^{7}}\right)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{x^{10} y^6}\).

Step 1: Simplify the inside of the parentheses

First, we need to simplify the expression inside the parentheses. The expression is \(\frac{x^{-2} y^{4} z^{0}}{x^{3} y^{7}}\). We know \(z^{0} = 1\), so \(z^0\) can be removed from the expression. This gives us \(\frac{x^{-2} y^{4}}{x^{3} y^{7}}\).

Step 2: Simplify the fraction using properties of exponents

Using the property of exponents, \(\frac{a^m}{a^n} = a^{m-n}\), we simplify each component of the fraction separately. For \(x\), we have \(x^{-2-3} = x^{-5}\). For \(y\), we have \(y^{4-7} = y^{-3}\). So the simplified expression inside the parentheses is \(x^{-5} y^{-3}\).

Step 3: Apply the power of a power property

The expression left is \((x^{-5} y^{-3})^2\). Apply the property \((a^m)^n = a^{mn}\) to each part: \((x^{-5})^2 = x^{-10}\) and \((y^{-3})^2 = y^{-6}\). Thus, the expression becomes \(x^{-10} y^{-6}\).

Step 4: Rewrite using positive exponents

Convert negative exponents to positive by using the property \(a^{-m} = \frac{1}{a^m}\). Therefore, \(x^{-10} = \frac{1}{x^{10}}\) and \(y^{-6} = \frac{1}{y^6}\). This results in \(\frac{1}{x^{10} y^6}\) as the final simplified expression using positive exponents.

Key Concepts

Simplifying Expressions

When simplifying expressions, the goal is to rewrite the expression in its simplest form.
This means removing unnecessary components and presenting them in a more straightforward manner. In algebra, expressions can include variables, coefficients, and exponents. Simplifying these expressions often involves combining like terms and using exponent rules.
In the given exercise, the expression inside the parentheses \[ \frac{x^{-2} y^{4} z^{0}}{x^{3} y^{7}} \] can be made simpler by removing any unnecessary parts.

Since any number or variable raised to the power of zero equals 1, we know that \( z^0 = 1 \).

This allows us to drop this term, simplifying our expression to \( \frac{x^{-2} y^{4}}{x^{3} y^{7}} \).

Next, by applying the properties of exponents, you can make this expression even simpler, as we'll explore in the next section.

Negative Exponents

A negative exponent is a way to represent the reciprocal of the base raised to the corresponding positive exponent.
In simpler terms, if you have an expression like \(a^{-m}\), it essentially means the reciprocal: \(\frac{1}{a^m}\).Understanding how to handle negative exponents is crucial when simplifying expressions. Let’s consider the expression after the zero power term is removed:\[ \frac{x^{-2} y^{4}}{x^{3} y^{7}}\]

For the variable \(x\), subtract the exponents: \(-2 - 3 = -5\), which gives \(x^{-5}\).

For \(y\), do the same: \(4 - 7 = -3\), resulting in \(y^{-3}\).

Thus, the expression now looks like \(x^{-5} y^{-3}\). In this form, both the \(x\) and \(y\) terms have negative exponents.

Power of a Power Property

This property states that when you raise a power to another power, you multiply the exponents.
It can be summarized with the formula: \[(a^m)^n = a^{mn}\]Using this rule helps in stepping to the next part of simplifying expressions.Applying this to our simplified expression \((x^{-5} y^{-3})^2\):

For \(x\), you apply the power of 2: \((x^{-5})^2 = x^{-10}\).

For \(y\), you do the same: \((y^{-3})^2 = y^{-6}\).

This leads to \(x^{-10} y^{-6}\).
Now, it’s important to rewrite the expression without negative exponents. We use the property \(a^{-m} = \frac{1}{a^m}\) to convert them:

\(x^{-10}\) becomes \(\frac{1}{x^{10}}\)

\(y^{-6}\) becomes \(\frac{1}{y^6}\)

Thus, your final expression using only positive exponents is:\[\frac{1}{x^{10} y^6}\]This transformation allows the expression to be presented in a clean and understandable manner, ready for further analysis or practical use.