Problem 15 - Algebra 1

Question

Use the quotient of powers property to simplify the expression. $$ \left(\frac{5}{4}\right)^{-3} $$

Step-by-Step Solution

Answer

The simplified form of \(\left(\frac{5}{4}\right)^{-3}\) is \(\frac{64}{125}\).

Step 1: Interpret Negative Exponent

From the rules of exponentiation, a number with a negative exponent is equal to the reciprocal of the number with a positive exponent. This means, \(-a = \frac{1}{a}\). Therefore, \(\left(\frac{5}{4}\right)^{-3}\) would become \(\left(\frac{4}{5}\right)^{3}\). Keep in mind that this step only flips the fraction and doesn't have anything to do with the quotient of powers yet.

Step 2: Apply Power to Numerator and Denominator

The rule involving exponents and division (\(\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}\)) says that a power on a quotient can be distributed to the numerator and the denominator of the quotient. We apply the exponent to each part of the fraction: \(\left(\frac{4}{5}\right)^{3} = \frac{4^3}{5^3}\).

Step 3: Simplify Numerator and Denominator

Calculate cube of numerator and denominator: \(\frac{4^3}{5^3} = \frac{64}{125}\). No more simplification is needed for this expression, because 64 and 125 share no common factors other than 1.

Key Concepts

Quotient of Powers Property

Understanding the quotient of powers property is a key skill in algebra. This property states that when you divide two powers with the same base, you can subtract the exponents. Mathematically, this is expressed as \[\[\begin{align*}\dfrac{a^m}{a^n} &= a^{m-n},\end{align*}\]\]where both the numerator and denominator have the same base 'a'. Essentially, division is turned into a much simpler subtraction problem. For positive exponents, this is straightforward, but when working with negative exponents, as with \[\[\begin{align*}\left(\dfrac{5}{4}\right)^{-3},\end{align*}\]\]the process includes additional steps to first convert the negative exponents into positive ones before applying the property.

In our exercise, we start with \[\[\begin{align*}\left(\dfrac{5}{4}\right)^{-3},\end{align*}\]\]but since we have a fraction to a negative power, we first convert that to \[\[\begin{align*}\left(\dfrac{4}{5}\right)^{3}\end{align*}\]\]by recognizing that a negative exponent indicates that we should take the reciprocal of the base.

Negative Exponents

Negative exponents can be a confusing topic, but the rule is quite simple. When you have an expression such as \[\[\begin{align*}a^{-n},\end{align*}\]\]where 'n' is a positive integer, it denotes the reciprocal, or one divided by the base raised to the positive exponent, \[\[\begin{align*}\dfrac{1}{a^n}.\end{align*}\]\]The idea is to transform the expression so that no negative exponents remain. To do so, we change the expression to its reciprocal form with a positive exponent. For the given exercise, we applied this rule to transform \[\[\begin{align*}\left(\dfrac{5}{4}\right)^{-3}\end{align*}\]\]to \[\[\begin{align*}\left(\dfrac{4}{5}\right)^3.\end{align*}\]\]This turns a potentially tricky expression into a more familiar and workable form. Remember, negative exponents don't make the number negative; they merely flip the fraction.

Exponents and Division

When it comes to exponents and division, the rules are designed to make calculations easier. For a fraction raised to an exponent, such as \[\[\begin{align*}\left(\dfrac{a}{b}\right)^n,\end{align*}\]\]the exponent 'n' applies to both the numerator 'a' and the denominator 'b' individually. This means you can rewrite the expression as \[\[\begin{align*}\dfrac{a^n}{b^n}.\end{align*}\]\]Using our initial exercise, after applying the negative exponent rule, we arrived at \[\[\begin{align*}\left(\dfrac{4}{5}\right)^{3},\end{align*}\]\]which then gets broken down into \[\[\begin{align*}\dfrac{4^3}{5^3} = \dfrac{64}{125}.\end{align*}\]\]This is a crucial step in the simplification process, facilitating the handling of each part of the fraction separately. The concept of distributing the exponent makes the process of dividing numbers by their powers much more straightforward.