Problem 19 - Algebra 1

Question

Use the power of a product property to simplify the expression. $$ \left(x^{3} y^{5}\right)^{4} $$

Step-by-Step Solution

Answer

The simplified form of \((x^{3} y^{5})^{4}\) is \( x^{12} * y^{20} \)

Step 1: Identify the base and exponent

First Identify the base and exponent in the expression \((x^{3} y^{5})^{4}\). Here, \(x^{3} y^{5}\) is the base within the parenthesis and 4 is the outer exponent.

Step 2: Apply the power of a product rule

Next, apply the power of a product rule or distributive law of exponents which states that \((ab)^n = a^n * b^n \). Here, \(a\) is \(x^{3}\), \(b\) is \(y^{5}\) and \(n\) is 4. So, \((x^{3} y^{5})^{4}\) can be written as \((x^{3})^{4} * (y^{5})^{4}\).

Step 3: Simplify using power to a power rule

Finally, simplify \((x^{3})^{4} * (y^{5})^{4}\) using the power to a power rule which states \((a^n)^m = a^{n*m} \). So, you get \(x^{3*4} * y^{5*4}\) which evaluates to \(x^{12} * y^{20}\).

Key Concepts

Understanding Exponents

Exponents are a fundamental concept in algebra. They represent how many times a number, known as the base, is multiplied by itself. For instance, in the expression \(x^3\), \(x\) is the base, and 3 is the exponent, which means \(x\) is used as a factor three times: \(x \times x \times x\).

• Using exponents allows for a concise way to write repeated multiplication.
• It simplifies working with very large or very small numbers by reducing them to more manageable forms.
• Understanding how to manipulate exponents is essential for solving various algebraic equations and expressions.

In this particular exercise, we deal with expressions that include not only simple exponents but powers of these exponents, which is where further rules come into play.

The Power to a Power Rule

The power to a power rule is one of the essential exponent properties in algebra. This rule states that if you have an exponent raised to another exponent, like \((a^n)^m\), you simplify it by multiplying the exponents: \(a^{n \cdot m}\).

• This helps simplify expressions where different powers are applied to the same base.
• It reduces complex expressions to simpler forms and makes calculations easier.

For the given exercise, applying the power to a power rule to \((x^3)^4\) resulted in \(x^{12}\), and similarly, \((y^5)^4\) became \(y^{20}\).

This technique streamlines the process of working with high-order exponents, making it a powerful tool in algebra.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing them into their simplest form for easier manipulation and understanding. This often includes combining like terms and applying exponent rules like the power of a product property and the power to a power rule.

• Start by identifying the underlying operations and their order, such as powers and products.
• Use properties of exponents such as distribution over multiplication to simplify complex expressions.
• Ensure all steps adhere to the mathematical principles governing exponents and products.

In the exercise, simplifying \((x^{3} y^{5})^{4}\) involved breaking it down into \((x^{3})^{4} * (y^{5})^{4}\) using the power of a product rule, then simplifying each factor with the power to a power rule. The result was the simplified expression \(x^{12} * y^{20}\).

Mastering these techniques paves the way for efficiently tackling more complex algebraic problems.