Problem 14 - Algebra 1: Concepts and Skills

Question

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

Step-by-Step Solution

Answer

The simplified form of \(\left(\frac{5}{4}\right)^{-3}\) using the power of a quotient is \(\frac{64}{125}\)

Step 1: Understand negative exponent

Firstly, understand how the negative exponent works. It implies taking the reciprocal of the base. So, \(\left(\frac{5}{4}\right)^{-3}\) becomes \(\left(\frac{4}{5}\right)^{3}\)

Step 2: Apply power of a quotient

Next, apply the power of a quotient rule. This implies that \(\left(\frac{4}{5}\right)^{3}\) becomes \(\left(\frac{4^{3}}{5^{3}}\right)\)

Step 3: Calculate the powers

Finally, do the calculations: \(4^{3} = 64\) and \(5^{3} = 125\). So \(\left(\frac{4^{3}}{5^{3}}\right)\) equals \(\left(\frac{64}{125}\right)\)

Key Concepts

Understanding Negative Exponents

When dealing with negative exponents, a common rule to remember is that they indicate the reciprocal of the base raised to the absolute value of the exponent.
For instance, if you have an expression like \left(\frac{5}{4}\right)^{-3}, the negative exponent \-3 suggests that we should flip the fraction \(\frac{5}{4}\) to \(\frac{4}{5}\) and then raise it to the power of 3.

• Negative exponents essentially "flip" the base.
• They mean "1 divided by" the base raised to the positive exponent.

Understanding this can help simplify expressions quite significantly.

Reciprocal and Its Role in Simplifying Expressions

The term reciprocal can be simply understood as flipping a fraction.
For any fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\).
This concept is very important when dealing with negative exponents.

• Reciprocals are crucial for converting negative exponents to positive ones.
• After converting with reciprocals, arithmetic becomes much simpler.

For example, converting \(\frac{5}{4}^{-3}\) to \(\frac{4}{5}^{3}\) makes it easier to work with the exponents in subsequent steps.

Application of the Power Rule

The power rule is a widely-used property in algebra, especially in simplifying expressions.
This rule says that when you raise a power to another power, you'll multiply the exponents.
However, when dealing with fractions or quotients, the power rule differs slightly but remains extremely useful.

• The power of a quotient means you raise both the numerator and denominator to the given power.
• In our example: \(\left(\frac{4}{5}\right)^{3} = \left(\frac{4^{3}}{5^{3}}\right)\).

This ensures that both parts of the fraction are raised to the power consistently, simplifying computation.

Simplifying Expressions by Calculating Powers

After applying the rules of reciprocals and power of a quotient, you arrive at a stage where arithmetic calculation takes place.
Calculating the powers means you need to know how to compute each part separately.

• Calculate \(4^3 = 64\).
• Calculate \(5^3 = 125\).

Thus, the original expression becomes \(\left(\frac{64}{125}\right)\).Finalizing the simplification means you have rewritten the expression in a form that is easy to understand.
It is a simple practice of breaking down the task into manageable steps.