Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-10,\) write each expression as a rational number without an exponent. $$ \frac{(2 \cdot 5)^{-4}}{5^{-2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{1}{400} \).

Step 1: Simplify the Expression

First, let's simplify the expression inside the fraction. We have \((2 \cdot 5)^{-4}\) in the numerator. This can be expanded to \(2^{-4} \times 5^{-4}\). So now the expression becomes: \[\frac{2^{-4} \times 5^{-4}}{5^{-2}} \]

Step 2: Simplify the Denominator

The denominator is \(5^{-2}\). To make it easier to deal with, let's move this term to the numerator by changing its exponent sign, according to the rule of negative exponents: \[\frac{2^{-4} \times 5^{-4} \times 5^2}{1} \]

Step 3: Combine Exponent Terms

Now, combine like terms in the numerator. The bases with the same number can be added together by adding their exponents. So we have: - For base \(5\), \(-4\) and \(+2\): \[ 5^{-4 + 2} = 5^{-2} \]Now the expression simplifies to:\[2^{-4} \times 5^{-2}\]

Step 4: Convert to a Rational Number

Convert the exponents into fractions by writing the numbers in the form of \(\frac{1}{x^n}\) where \(n\) is positive:\[\frac{1}{2^4} \times \frac{1}{5^2} = \frac{1}{16} \times \frac{1}{25} = \frac{1}{400}\]

Step 5: Final Result

The expression \( \frac{(2 \cdot 5)^{-4}}{5^{-2}} \) simplifies to the rational number \( \frac{1}{400} \).

Key Concepts

Rational Number

A rational number is any number that can be expressed as the quotient or fraction of two integers, where the numerator is an integer and the denominator is a non-zero integer. This means any whole number, fraction, or terminating decimal fits the description. Rational numbers are characterized by their ability to be written in the form \( \frac{a}{b} \), where \(a\) and \(b\) are integers and \(b eq 0\). Here, the term ‘rational’ comes from ‘ratio’, highlighting their nature of expressing ratios between integers. For example:

• \( \frac{3}{4} \) is a rational number.
• \(-5\) can be written as \(\frac{-5}{1}\).
• \(0.75\), which converts to \(\frac{3}{4}\), is rational as well.

The expression \( \frac{1}{400} \) is a rational number since it fits the definition—both the numerator and the denominator are integers. Whenever you're working with fractions or expressions, converting them into rational numbers can make them easier to understand and manipulate. This conversion often involves simplifying the fraction or changing its form by adjusting exponents.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its smallest form. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD). A simplified fraction makes it easier to understand and compare with other numbers.Here are the steps to simplify a fraction:

• Find the GCD of the numerator and the denominator.
• Divide both the numerator and denominator by the GCD.
• Rewrite the fraction in its simplest form.

Simplifying fractions eliminates complexity, making further calculations straightforward. In expressions involving exponents, like \( \frac{2^{-4} \times 5^{-2}}{1} \), simplification often involves converting each term to its simplest form first, as seen in the original solution:- Here, each term with a negative exponent is converted to a positive one by placing it under 1 (flipping from the numerator to the denominator or vice versa).- Once expressed as \( \frac{1}{16} \times \frac{1}{25} \), multiplying them results in \( \frac{1}{400} \), a simplified rational number.

Exponent Rules

Exponent rules are guidelines that simplify calculations involving powers of numbers or variables. Understanding these rules is key to working efficiently with expressions like the one in the exercise.Key exponent rules include:

• Product of Powers: When multiplying like bases, add their exponents. For example, \(a^m \times a^n = a^{m+n}\).
• Power of a Power: When raising an exponentiated number to another power, multiply the exponents. That is, \((a^m)^n = a^{m \times n}\).
• Quotient of Powers: When dividing like bases, subtract their exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• Negative Exponents: A negative exponent indicates a reciprocal: \(a^{-n} = \frac{1}{a^n}\).

In the exercise, these rules were utilized extensively:- The negative exponents \(5^{-4}\) and \(5^{-2}\) were adjusted using rules of negative exponents and quotient of powers.- After simplifying each part of the fraction, the expression became easier to work with, eventually leading to the final answer. Recognizing how exponents play a role in calculations helps frame solving steps logically and efficiently.