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Chapter 9: Problem 36

For Exercises, simplify. $$\frac{6 y^{8}}{8 y^{9}}$$

Short Answer

Expert verified
The simplified form of the given expression is \(\frac{3}{4 y}\)

Step by step solution

01

Simplify Coefficients

Start by simplifying the coefficients 6 and 8 separately by finding the greatest common divisor which is 2. This simplifies down the fraction to \(\frac{3}{4}\).
02

Simplify Exponential Part

Next, we need to simplify the variable exponent part of the expression, \(y^{8} / y^{9}\). According to the exponent division rule, we subtract the exponent of the denominator from that of the numerator. This leaves us with \(y^{8-9} = y^{-1}\).
03

Combine Simplified Parts

Finally, we combine the simplified coefficients and the simplified exponential part: \(\frac{3}{4} * y^{-1}\). Keep in mind that \(y^{-1}\) is equivalent to \(1/ y\), so the expression can also be written as \(\frac{3}{4 y}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents
Exponents are a way to express repeated multiplication of a number by itself. For example, in the expression \( y^8 \), \( y \) is multiplied by itself 8 times. When dividing two expressions with the same base, such as \( y^8 \) and \( y^9 \), we use the rule \( a^m / a^n = a^{m-n} \), which says to subtract the exponent in the denominator from the exponent in the numerator. So, \( y^8 / y^9 = y^{8-9} = y^{-1} \). A negative exponent, like \( y^{-1} \), means we take the reciprocal, turning it into \( 1/y \). This is a key idea to remember when working with exponents.
Coefficients
Coefficients are the numerical parts of terms in an expression. In our example, \( 6 \) and \( 8 \) are coefficients. Simplifying coefficients involves finding the greatest common divisor (GCD). The GCD of 6 and 8 is 2. Dividing both by 2, we simplify the fraction \( \frac{6}{8} \) to \( \frac{3}{4} \). This simplification helps make expressions more concise and easier to work with, particularly when combined with other components like exponents.
Division of Variables
Dividing variables involves simplifying expressions with the same base but different exponents. When you divide \( y^8 \) by \( y^9 \), you're effectively looking to subtract their exponents: \( 8 - 9 \) resulting in \( y^{-1} \). You end up with the reciprocal of \( y \), which is \( 1/y \). Remembering the rule of subtracting exponents is handy: it's a simple operation but forms the base for understanding more complex algebraic expressions.
Simplifying Fractions
Simplifying fractions involves reducing them to their simplest terms. This means ensuring the numerator and the denominator do not have any common factors other than 1. In our example, the fraction \( \frac{6}{8} \) was simplified to \( \frac{3}{4} \) by dividing both terms by their greatest common divisor, which is 2. When dealing with fractions involving variables, ensure any shared factors in both the numerator and the denominator are simplified as well. This makes expressions cleaner and reduces potential errors in further calculations.

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