Problem 3 - Algebra 1

Question

Use the quotient of powers property to simplify the expression. $$ \frac{5^{4}}{5^{1}} $$

Step-by-Step Solution

Answer

The simplified expression is 125.

Step 1: Identify the Quotient of Powers Property

The quotient of powers property states that for any non-zero numbers and integers m and n, \(a^{m}/a^{n} = a^{m-n}\). This rule can be applied here because both terms in the expression have the same base, which is the number 5.

Step 2: Apply the Quotient of Powers Property

Apply the quotient of powers property to simplify the expression. Consequently, the expression \(5^{4}/5^{1}\) becomes \(5^{4 - 1}\), because 4 is subtracted by 1, which is the operation indicated by the quotient of powers property.

Step 3: Simplify the Resulting Expression

Simplify the resulting expression, \(5^{4 - 1}\) becomes \(5^{3}\).

Step 4: Calculate the final Answer

Calculate the final answer, \(5^{3}\) results into 125.

Key Concepts

Quotient of Powers

The quotient of powers property is a fundamental rule in algebra that simplifies expressions where exponents with the same base are divided. This property is especially useful when working with powers of a number in fractional form. If you have a term in the form \( \frac{a^m}{a^n} \), the rule states that you can subtract the exponent in the denominator from the exponent in the numerator. This gives you \( a^{m-n} \). Here are key points about this property:

• The bases in the numerator and denominator must be the same.
• Both exponents, \( m \) and \( n \), must be integers.
• It requires that the base \( a \) is not zero, otherwise the division would be undefined.

Understanding and applying the quotient of powers property can greatly simplify complex fractional expressions, making them easier to solve and understand.

Simplifying Expressions

Simplifying expressions is a critical skill in mathematics that involves reducing an expression to a simpler form while still maintaining its original value. In this context, it involves applying algebraic properties, like the quotient of powers property, to remove unnecessary complexity.

When simplifying \( \frac{5^4}{5^1} \), for example, you apply the quotient of powers property to subtract the exponent in the denominator from the exponent in the numerator, resulting in \( 5^{4-1} \) which simplifies to \( 5^3 \).

This simplified expression is much easier to manage or further manipulate if necessary. Here are a few tips for simplifying expressions:

• Identify common bases and apply exponent rules appropriately.
• Look for opportunities to reduce fractions or expressions to their simplest forms.
• Maintain the integrity of the expression’s value throughout the simplification process.

Powers of a Number

Understanding powers of a number is crucial in mastering many algebraic concepts. A power of a number means multiplying that number by itself a certain number of times. When expressed as \( a^n \), it signifies the base \( a \) multiplied by itself \( n \) times. For instance, \( 5^3 \) means 5 multiplied by itself twice more: \( 5 \times 5 \times 5 = 125 \).

Working with powers is essential not only for simplification but also for solving equations and modeling real-world situations. Key facts about powers include:

• \( a^0 \) equals 1 for any nonzero \( a \).
• Multiplying powers with the same base adds their exponents: \( a^m \times a^n = a^{m+n} \).
• The inverse operation, roots, can also be seen as fractional exponents.

Mastering the concept of powers allows for more efficient problem-solving and helps in understanding exponential growth and decay, amongst other applications in math and science.