Problem 83 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{50 b^{10}}{70 b^{5}} $$

Step-by-Step Solution

Answer

\( \frac{5b^{5}}{7} \)

Step 1: Identify the Fractions

You have the expression \( \frac{50 b^{10}}{70 b^{5}} \). This is a fraction where the numerator is \( 50b^{10} \) and the denominator is \( 70b^{5} \).

Step 2: Simplify the Numerical Coefficients

Focus on the coefficients 50 and 70. Both can be simplified by their greatest common divisor, which is 10: \[ \frac{50}{70} = \frac{5}{7} \]. Thus, the fraction becomes \( \frac{5b^{10}}{7b^{5}} \).

Step 3: Apply the Laws of Exponents

For the exponents part, subtract the exponents of \( b \) in the denominator from the exponents in the numerator as per the law \( \frac{b^m}{b^n} = b^{m-n} \). So, \( b^{10-5} = b^{5} \).

Step 4: Rewrite the Simplified Expression

Combine the simplified coefficient and the exponent: \( \frac{5b^{5}}{7} \).

Key Concepts

Exponents in Algebraic Expressions

Exponents are a shorthand way to represent repeated multiplication of a number by itself. In the expression \( b^{10} \), the base is \( b \) and the exponent is 10, indicating that \( b \) is multiplied by itself 9 more times, totaling 10. This concept is vital when simplifying algebraic expressions, as it allows us to condense lengthy expressions into a compact form. Applying the rules of exponents can significantly simplify calculations. For example, when dividing terms with the same base like \( b^{10} \) by \( b^5 \), we subtract the exponents: \( b^{10-5} = b^5 \). This rule is rooted in the property that \( b^m / b^n = b^{m-n} \), making operations easier and tidier. Understanding this concept helps simplify expressions quickly and efficiently.

Understanding Fractions

Fractions represent parts of a whole and consist of two parts: the numerator and the denominator. In the fraction \( \frac{50b^{10}}{70b^5} \), the numerator is \( 50b^{10} \), and the denominator is \( 70b^5 \). Fractions can be simplified by reducing the terms in the numerator and denominator by their common factors, which makes calculations easier. Simplifying fractions is like finding the simplest form that fully represents the proportion. By understanding how fractions function, simplifying them in various algebraic contexts becomes a valuable skill.

The Greatest Common Divisor

The greatest common divisor (GCD), sometimes called the greatest common factor, is the largest number that divides two or more numbers without leaving a remainder. Finding the GCD is crucial when simplifying fractions, as it helps to reduce them to their lowest terms. For example, when simplifying \( \frac{50}{70} \), the GCD of 50 and 70 is 10. By dividing both the numerator and the denominator by 10, we achieve \( \frac{5}{7} \). This process simplifies the fraction to its simplest form, making it easier to work with and understand.

Numerator and Denominator Explained

In any given fraction, the numerator is the top part, while the denominator is the bottom part. The numerator represents how many parts of that fraction are being considered, whereas the denominator indicates the total number of equal parts.Using the fraction \( \frac{50b^{10}}{70b^5} \) as an example:

• The numerator is \( 50b^{10} \), illustrating the portion of the division symbol \( \div \) that represents the dividend.
• The denominator is \( 70b^5 \), representing the divisor in the division relationship.

Understanding these roles within a fraction is key in manipulating and simplifying algebraic expressions accurately. Recognizing these parts also aids in applying other processes such as finding the GCD or working across differing mathematical operations like multiplication and division that apply to fractions.