Problem 17 - Algebra 1

Question

Use the quotient of powers property to simplify the expression. $$ \left(\frac{x^{3}}{y^{5}}\right)^{6} $$

Step-by-Step Solution

Answer

The simplified form of \( \left(\frac{x^{3}}{y^{5}}\right)^{6} \) is \( x^{18} / y^{30} \).

Step 1: Identify the Expression

The problem involves the expression \(\left(\frac{x^{3}}{y^{5}}\right)^{6}\). This is an expression involving powers and the quotient operation.

Step 2: Apply the Quotient of Powers Property

We apply the quotient of powers property which states that \( (a/b)^n = a^n / b^n \). Using this property, the given expression simplifies to \( (x^{3})^{6} / (y^{5})^{6} \).

Step 3: Simplify the Expression

Now apply the rule of exponents which states that \( (a^n)^m = a^{n*m} \). The expression simplifies to \( x^{3*6} / y^{5*6} \), which further simplifies to \( x^{18} / y^{30} \).

Key Concepts

Quotient of Powers

The concept of the quotient of powers is fundamental when dealing with expressions involving exponents. In the expression given, the quotient of powers property is crucial in simplifying the base terms to their respective powers. The property can be stated as:

• When you have a division of two like bases with exponents, you keep the base the same and subtract the exponents. In typical practice, however, the property is stated for an entire fraction being raised to a power, which means: \( (\frac{a}{b})^n = \frac{a^n}{b^n} \).

This allows us to separate the fraction and apply the exponent individually to both the numerator (top part) and denominator (bottom part). In our original expression, we applied \(((x^{3})^6 / (y^{5})^6\)). Each term retains its base, while each exponent becomes multiplied by the power they are raised to.

Simplifying Expressions

Simplifying expressions involves using the rules of arithmetic and algebra to reduce an expression into a simpler or alternate form without changing its value. In our step-by-step solution, we use the quotient of powers to split the power over the fraction:

• Split the power to apply separately over numerator and denominator (\((x^{3})^{6} / (y^{5})^{6}\)).
• Multiply the exponents: \((x^{18}/ y^{30})\).

This results in expressions which are often easier to comprehend and use in further calculations. In general, act as if you're peeling layers of an onion, removing complexity while keeping the onion itself—just in easier forms to further work with.

Exponent Multiplication

Exponent multiplication is a technique that comes into play when you have expressions such as \((a^n)^m\). The rule to remember is that you multiply the exponents together:

• When \(a^n\) is raised to another power \(m\), it simplifies to \(a^{n \times m}\).
• Use this when an exponent is itself being raised to another power.

In our expression, `x` is raised to the power of 3 and then this whole term is further raised to the power of 6, which results in \(x^{3 \times 6} = x^{18}\). Similarly, \(y^{5}\) being raised to the power of 6 results in \(y^{5 \times 6} = y^{30}\). This simplifies the expression greatly, turning many layers of exponential complexity into a straightforward multiplication.

Power of a Power Rule

The power of a power rule is one of the key rules of exponents which states that when a term with an exponent is raised to a further power, you simply multiply the exponents:

• If you have an expression \((a^n)^m\), it becomes \(a^{n \times m}\).

This rule allows us to handle nested powers efficiently. In our problem, observe that \((x^3)^6\) was effectively causing a multiplication of the exponent 3 by the outer power 6, leading to \(x^{18}\). Similarly, \((y^5)^6\) simplifies to \(y^{30}\). This routine application of power of a power principle transforms the expression significantly, allowing for efficient computation or further algebraic manipulation. It greatly reduces the complexity of multiplying large powers manually and aids in simplifying calculations.