Problem 27 - Algebra 1: Concepts and Skills

Question

Simplify the quotient. $$ \frac{(-3)^{6}}{(-3)^{6}} $$

Step-by-Step Solution

Answer

The simplified form of \(\frac{(-3)^{6}}{(-3)^{6}}\) is 1.

Step 1: Identify the Base and the Exponents

In this exercise, the base of the expression is -3. The exponent in the numerator is 6, and the exponent in the denominator is also 6.

Step 2: Apply the Quotient of Powers Property

The Quotient of Powers Property states that \((a^{m}) / (a^{n}) = a^{m-n}\). In this case, \(a = -3\), \(m = 6\), and \(n = 6\). Applying this property gives \((-3)^{6-6}\).

Step 3: Simplify the Exponent

Calculating \(6 - 6\) in the exponent gives \((-3)^{0}\).

Step 4: Apply the Zero Exponent Property

Finally, remember that any non-zero number raised to the power of 0 is 1. Therefore, \((-3)^{0}\) simplifies to 1.

Key Concepts

Zero Exponent Property

When working with exponents, the Zero Exponent Property is a key concept to understand. This property tells us that any non-zero number raised to the power of zero equals one. It might seem surprising at first, but this rule holds true for all non-zero numbers. Here’s how it works:

• For example, in the expression \((-3)^6/(-3)^6\), when you simplify it to \((-3)^0\), the zero exponent applies.
• The result is 1, because \((-3)^0 = 1\).
• This property helps simplify complex expressions by reducing them to simpler forms.

Using this property can make solving algebraic problems much quicker and easier.

Simplifying Exponents

Simplifying expressions with exponents can make calculations far more manageable. The Quotient of Powers Property is a useful tool here. It states that if you divide two powers with the same base, you simply subtract the exponents. Here’s how this looks in practice:

• In the expression \(\frac{(-3)^6}{(-3)^6}\), both bases are -3.
• Subtracting the exponents gives \((-3)^{6-6} = (-3)^0\).
• You then use the Zero Exponent Property to simplify \((-3)^0\) to 1.

Breaking down each step reduces the chance of error and leads to clearer understanding.

Algebraic Expressions

Algebraic expressions consist of variables, numbers, and operations. Mastery of exponents is essential for manipulating these expressions. By understanding principles like the Quotient of Powers and Zero Exponent Property, you can simplify expressions efficiently. Consider these points:

• Identify the different components such as the base and exponents.
• Use properties like exponent rules to rewrite the expression in simpler forms.
• Simplifying helps in solving equations and understanding relationships between variables.

Building a solid foundation with these concepts will enhance your ability to work through more complex algebra problems in the future.