Problem 72 - Algebra 2

Question

Simplify each expression. 125\(\cdot 125^{-\frac{1}{3}}\)

Step-by-Step Solution

Answer

The simplified form of the expression 125\cdot 125^{-\frac{1}{3}} is 25.

Step 1: Understand the negative exponent

A negative exponent means that we invert the base and make the exponent positive. Hence, \(125^{-\frac{1}{3}}\) becomes \(\frac{1}{125^\frac{1}{3}}\) .

Step 2: Simplify the cubic root

In the denominator, 125 raised to the power of 1/3 is equivalent to the cubic root of 125. The cubic root of 125 is 5. Hence, our expression becomes \(125 \cdot \frac{1}{5}\).

Step 3: Multiply

Multiply 125 by 1/5 to get 25.

Key Concepts

Negative Exponents

Negative exponents often appear intimidating at first, but understanding their meaning can simplify many math problems. Simply put, a negative exponent indicates that we need to take the reciprocal of the base and convert the exponent to a positive.
For example, in the expression \(125^{-\frac{1}{3}}\), the base is 125 and the exponent is \(-\frac{1}{3}\). When we convert it, we get \(\frac{1}{125^{\frac{1}{3}}}\).
Here's how it works:

• Take the reciprocal of the base: Which means we invert it. So, 125 becomes the denominator of a fraction.
• Convert the exponent to positive: Once the base is inverted, the negative exponent becomes positive, making calculations easier.

Understanding negative exponents is crucial as they frequently appear in various mathematical expressions and equations, especially when dealing with roots or simplifying fractions.

Cubic Roots

Cubic roots simplify expressions by reducing the exponentiation involved, specifically for cubes. When dealing with roots, it's essential to understand they're the inverse operation of exponentiation.
For the cubic root of a number, you're essentially asking what number multiplied by itself three times gives you that amount. The notation \(125^{\frac{1}{3}}\) represents the cubic root of 125.

• Identify the cubic root: In our specific problem, the cubic root of 125 is 5, because \(5^3 = 125\).
• Practical approach: If you're unsure about finding cubic roots by heart, consider breaking the number down into its prime factors. In this case, 125 is \(5 \times 5 \times 5\), leading you directly to the cubic root being 5.

Knowing how to find cubic roots is essential for simplifying equations involving them, making complex expressions more manageable.

Simplifying Expressions

Simplifying expressions is a fundamental skill in mathematics, allowing you to reduce complex formulas into more manageable forms. This process involves several steps, including applying rules of exponents, calculating roots, and performing basic arithmetic operations.
In our original problem, we faced the expression \(125 \cdot 125^{-\frac{1}{3}}\). Breaking it down:

• Apply negative exponent rules: Convert \(125^{-\frac{1}{3}}\) to \(\frac{1}{125^{\frac{1}{3}}}\).
• Simplify root: Recognize that \(125^{\frac{1}{3}} = 5\). Hence, the expression transitions to \(125 \cdot \frac{1}{5}\).
• Perform arithmetic: Calculate the multiplication \(125 \cdot \frac{1}{5}\) to get the final result, which is 25.

By systematically applying these steps, you can simplify expressions efficiently and accurately, a skill applicable across various sectors of math and science.