Problem 33 - Algebra A Combined Function

Question

Simplify each expression. Write each result using positive exponents only. $$ \left(x^{5} y^{3}\right)^{-3} $$

Step-by-Step Solution

Answer

\(\frac{1}{x^{15} y^9}\)

Step 1: Apply the Power of a Power Rule

The Power of a Power Rule states that when you raise a power to another power, you multiply the exponents. So, first, apply this rule to the expression \((x^5 y^3)^{-3}\). This results in \(x^{5 imes (-3)} y^{3 imes (-3)}\).

Step 2: Simplify the Exponents

Multiply the exponents inside the parentheses by -3: - For \(x\): \(5 \times (-3) = -15\)- For \(y\): \(3 \times (-3) = -9\)Thus, the expression becomes \(x^{-15} y^{-9}\).

Step 3: Rewrite with Positive Exponents

To express negative exponents as positive, recall that \(a^{-n} = \frac{1}{a^n}\). Apply this rule:- For \(x^{-15}\): \(x^{-15} = \frac{1}{x^{15}}\)- For \(y^{-9}\): \(y^{-9} = \frac{1}{y^9}\)Combine these to give \(\frac{1}{x^{15} y^9}\).

Key Concepts

Power of a Power Rule

The Power of a Power Rule is an important concept in algebra that helps to simplify expressions where one power is raised to another power. When you encounter such situations, simply multiply the inside exponent by the outside exponent. This results in a single power with a new exponent.

For example, when we simplify \((x^5 y^3)^{-3}\), we apply this rule by multiplying each exponent inside the parentheses by -3:

• For \(x\), multiply 5 by -3 to get -15
• For \(y\), multiply 3 by -3 to get -9

Now, you have \(x^{-15} y^{-9}\). This makes calculations more straightforward by simplifying large and complicated powers into a single exponent.

Negative Exponents

Dealing with negative exponents can seem tricky, but it's easier if you remember that they represent reciprocal powers. When you see a negative exponent, think of it as taking the reciprocal of the base raised to the corresponding positive exponent.

In our example, after applying the power of a power rule, we ended up with \(x^{-15}\) and \(y^{-9}\). Here's how you convert these into something more manageable:

• For \(x^{-15}\), it's equivalent to \(\frac{1}{x^{15}}\)
• For \(y^{-9}\), it's equivalent to \(\frac{1}{y^9}\)

This rule helps in simplifying expressions and keeping them in a standard positive exponent form, which is generally easier to work with.

Positive Exponents

A fundamental aspect of algebraic simplification is writing expressions with only positive exponents. Positive exponents are straightforward and indicate how many times a number (the base) is used as a factor.

In our task, filling in positive exponents involves transforming \(x^{-15} y^{-9}\) into its reciprocal form. By applying the relationship that \(a^{-n} = \frac{1}{a^n}\), you rewrite:

• \((x^{-15})\) as \(\frac{1}{x^{15}}\)
• \((y^{-9})\) as \(\frac{1}{y^9}\)

Finally, combine them to arrive at \(\frac{1}{x^{15} y^9}\). This approach not only simplifies the expression but also makes it easier to interpret and use in further algebraic manipulations.