Problem 11 - Elementary Algebra

Question

Make use of either or both the power rule for products and the power rule for powers to simplify each expression. $$ \left(x^{3} x^{5} y^{2} y^{6}\right)^{9} $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression: $(x^3 \cdot x^5 \cdot y^2 \cdot y^6)^9$. Answer: The simplified expression is $x^{72} y^{72}$.

Step 1: Simplify the expression inside the parenthesis

In order to simplify the expression, we will use the power rule for products and combine terms with the same base. The expression inside the parenthesis is: $$ x^3 \cdot x^5 \cdot y^2 \cdot y^6 $$ Using the power rule for products to combine terms with the same base, we get: $$ (x^{3+5})(y^{2+6}) = x^8 y^8 $$

Step 2: Apply the power rule for powers

Now, we have the simplified expression inside the parenthesis and can apply the power rule for powers to raise it to the power of 9. The expression we have is: $$ (x^8 y^8)^9 $$ Using the power rule for powers, we get: $$ x^{8 \cdot 9} y^{8 \cdot 9} = x^{72} y^{72} $$

Step 3: Write the final simplified expression

After applying the power rule for products and the power rule for powers, we have obtained the final simplified expression: $$ x^{72} y^{72} $$

Key Concepts

Understanding the Power Rule

Exponentiation involves multiplying a number, called the base, by itself a certain number of times, known as the exponent. The power rule is a crucial principle in exponentiation that helps simplify expressions involving powers. There are primarily two power rules to consider.

The first is the power rule for products, which is the focus here. When multiplying powers that have the same base, you can simplify the expression by adding the exponents. For example:

• If you have \(x^a \times x^b\), it can be rewritten as \(x^{a+b}\).

This property helps reduce complex expressions into simpler forms.

The second rule is the power rule for powers. This rule is used when you have a power raised to another power. You can simplify it by multiplying the exponents:

• For instance, \((x^a)^b\) becomes \(x^{a \cdot b}\).

These rules transform complicated equations, making them much easier to manage.

Working with the Product of Powers

The product of powers involves expressions where multiple terms with the same base are multiplied together. Understanding this concept helps in making exponentiation calculations more intuitive.

In the original exercise, the expression \(x^3 \cdot x^5\) is a classic example. Both times, the base is \(x\), so you can apply the product of powers rule:

• The expression simplifies to \(x^{3+5} = x^8\).

The same goes for other terms with the same base. If you look at \(y^2 \cdot y^6\), by adding the exponents:

• It becomes \(y^{2+6} = y^8\).

This simplification is immensely powerful when dealing with complicated algebraic expressions. It consolidates multiple terms into a more manageable form.

Simplifying Expressions Efficiently

Once you have simplified the terms within parentheses using product rules, you may encounter expressions raised to additional powers, necessitating further simplification.

In this exercise, the simplified expression \((x^8 y^8)^9\) uses the power rule for powers to simplify further. This technique involves distributing the outside exponent across each term inside the parentheses:

• Multiply the exponent inside by the outside exponent for each term, giving \(x^{8 \cdot 9} = x^{72}\).
• Do the same for \(y\): \(y^{8 \cdot 9} = y^{72}\).

By understanding how to simplify using these rules efficiently, you transform potentially complex algebraic expressions into their simplest form, making calculations more straightforward. This aids not only in your calculations but also in your overall grasp of algebraic principles.