Problem 39 - Algebra 1

Question

Simplify the expression. The simplified expression should have no negative exponents. $$ \frac{\left(r^{3}\right)^{4}}{\left(r^{3}\right)^{8}} $$

Step-by-Step Solution

Answer

The simplified version of the given expression with no negative exponents is \( 1/r^{12} \).

Step 1: Apply the Power of a Power Rule

Applying the power of a power rule to the numerator and the denominator separately: in the numerator \( (r^{3})^{4} = r^{3*4} = r^{12} \) and in the denominator \( (r^{3})^{8} = r^{3*8} = r^{24} \). So, the original expression can be written as \( r^{12} / r^{24} \).

Step 2: Apply Exponent Subtraction Rule

Now apply the rule for exponent subtraction: \( a^n / a^m = a^{n-m} \), it means subtract the exponent of the denominator from the exponent of the numerator: \( r^{12-24} = r^{-12} \).

Step 3: Deal With Negative Exponents

Having a negative exponent might seem like a violation of the exercise requirements, but remember that \( a^{-n} = 1/a^n \). It means that exponents can be made positive by reciprocating the base. So, \( r^{-12} \) is equivalent to \( 1/r^{12} \).

Key Concepts

Power of a Power Rule

When simplifying expressions with exponents, the Power of a Power Rule is an essential concept to grasp. It states that when you have an exponent raised to another exponent, you multiply the exponents together. For example, when you see an expression like \( (x^n)^m \), you would apply this rule to obtain \( x^{n*m} \).

In the provided exercise, the expression \( (r^3)^4 \) in the numerator and \( (r^3)^8 \) in the denominator apply to this rule. By multiplying the exponents, we convert \( (r^3)^4 \) into \( r^{3*4} \) which simplifies to \( r^{12} \), and \( (r^3)^8 \) simplifies to \( r^{24} \). Mastery of this rule makes manipulating expressions with multiple exponents straightforward and helps in further simplification steps.

Exponent Subtraction Rule

Moving on to the Exponent Subtraction Rule, which comes into play when dividing like bases with exponents. If you have \( a^n / a^m \), where 'a' represents the base and 'n' and 'm' are the exponents, you subtract the exponent of the denominator from the exponent of the numerator, resulting in \( a^{n-m} \).

This is a key step in the provided exercise after applying the Power of a Power Rule. We have \( r^{12} / r^{24} \) which is an ideal scenario for the Exponent Subtraction Rule. Subtracting the two exponents, 12 and 24, gives us \( r^{-12} \). Understanding and applying this rule correctly is crucial to simplifying expressions of this nature.

Negative Exponents

The concept of Negative Exponents might be challenging at first, but it's quite simple once you get the hang of it. A negative exponent indicates that the base number is on the wrong side of a fraction and should be reciprocated. Essentially, \( a^{-n} \) is equal to \( 1/a^n \). In the given exercise, we encountered \( r^{-12} \) after applying the Exponent Subtraction Rule. To make this exponent positive and adhere to the exercise requirements, we take the reciprocal of the base, which turns \( r^{-12} \) into \( 1/r^{12} \).

Knowing how to handle negative exponents is a fundamental skill, as it ensures expressions are simplified correctly and all exponents are positive, as often necessitated by mathematical conventions and the requirements of a given problem.