Problem 67 - Intermediate Algebra

Question

Simplify each expression. All variables represent positive real numbers. See Example 7. $$ 125^{-1 / 3} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{5}\).

Step 1: Rewrite as a Radical

The given expression is \(125^{-1/3}\). To simplify this, first express the negative power as a reciprocal. We get \((125^{1/3})^{-1}\) which is the same as \(\frac{1}{125^{1/3}}\). Now, note that \(125^{1/3}\) indicates the cube root of 125.

Step 2: Calculate the Cube Root

Determine \(125^{1/3}\) which is the cube root of 125. Identifying the cube root means finding a number that, when multiplied by itself three times, equals 125. This number is 5 because \(5^3 = 125\). Therefore, \(125^{1/3} = 5\).

Step 3: Simplify the Expression

Now substitute back the value of the cube root into the expression. Thus, \(\frac{1}{125^{1/3}} = \frac{1}{5}\). This is the simplified form of the expression.

Key Concepts

Understanding Cube Roots

A cube root is a number that, when used three times in a multiplication, gives the original number. For the expression \( 125^{-1/3} \), you first need to identify the cube root of 125. The cube root of a number \( n \) is written as \( n^{1/3} \). This is because raising a number to the power of 1/3 is the same as finding the cube root. In this case, since \( 5^3 = 125 \), the cube root of 125 is 5.
To simplify \( 125^{1/3} \), you look for a number that multiplied by itself three times results in 125. Knowing \( 5 \times 5 \times 5 = 125 \) helps you understand that the cube root of 125 is simply 5. Simplifying cube roots is a straightforward process once you understand this basic multiplication.

Exploring Negative Exponents

Negative exponents can be a bit confusing at first, but they are simply a way to denote reciprocals. When you have a negative exponent, it means that instead of multiplying by that base, you take one over that multiplication. For example, \( a^{-n} = \frac{1}{a^n} \).
This rule is key when you're faced with expressions like \( 125^{-1/3} \). You convert this to \( \frac{1}{125^{1/3}} \). Notice how the negative exponent becomes a reciprocal. It's just flipping the base into the denominator. Understanding that negative exponents mean taking the reciprocal allows you to handle expressions more confidently and simplifies complex problems into simpler tasks.

The Role of Reciprocals in Simplification

Reciprocals are essential in mathematical simplifications, especially when dealing with negative exponents. A reciprocal is simply one divided by the number. For instance, the reciprocal of a number \( x \) is \( \frac{1}{x} \).
When you substitute the cube root of 125 into the expression \( \frac{1}{125^{1/3}} \), it simplifies to \( \frac{1}{5} \). This step shows you how reciprocals help in handling complex expressions by flipping the number after dealing with the negative exponent.

• If you understand reciprocals, simplification becomes a lot faster and less confusing.
• Using the reciprocal is often the last step in many simplification processes.

Grasping reciprocals help you manage expressions involving both negative exponents and roots more smoothly.