Problem 68 - Algebra 2

Question

Simplify each expression. Assume that all variables are positive. $$y^{\frac{5}{7}} \div y^{\frac{3}{4}}$$

Step-by-Step Solution

Answer

The simplified expression of \(y^{\frac{5}{7}} \div y^{\frac{3}{4}}\) is \(\frac{1}{y^{\frac{1}{28}}}\).

Step 1: Identify the given exponents

The expression is \(y^{\frac{5}{7}} \div y^{\frac{3}{4}}\). Here, the exponent of the numerator is \(\frac{5}{7}\), and the exponent of the denominator is \(\frac{3}{4}\). The bases in both terms are same, which is \(y\). So, the expression fits the form \(a^m/a^n = a^{m-n}\).

Step 2: Subtract the exponents

To simplify the provided expression, subtract the exponent of the denominator from the exponent of the numerator: \[y^{\frac{5}{7} - \frac{3}{4}}\]. This involves subtraction of fractions.

Step 3: Subtraction of fractions

Before subtract the fractions, find a common denominator. The least common denominator of 7 and 4 is 28. Rewrite both fractions with the common denominator. Then do the subtraction: \(\frac{5 \times 4}{7 \times 4} - \frac{3 \times 7}{4 \times 7} = \frac{20}{28} - \frac{21}{28} = \frac{-1}{28}\).

Step 4: Final Simplification

The simplified expression becomes \(y^{\frac{-1}{28}}\). According to the rules of exponents, this is the same as \(\frac{1}{y^{\frac{1}{28}}}\). The expression is now simplified.

Key Concepts

Fraction Subtraction

Fraction subtraction might sound intimidating, but it's actually quite straightforward. Imagine you have fractions that you need to subtract, such as those involved in our exercise. To perform subtraction:

• First, identify a common denominator. The common denominator allows you to compare and subtract fractions more easily.
• For example, if faced with fractions like \(\frac{5}{7}\) and \(\frac{3}{4}\), find the smallest number that both denominators (7 and 4) can divide into, called the least common denominator (LCD). Here, it's 28.
• Next, convert each fraction to have this common denominator: \(\frac{5}{7}\) becomes \(\frac{20}{28}\) and \(\frac{3}{4}\) becomes \(\frac{21}{28}\).
• Finally, subtract the numerators while keeping the common denominator: \(\frac{20}{28} - \frac{21}{28} = \frac{-1}{28}\).

Subtracting fractions becomes much simpler with these steps, turning potentially complex operations into clear processes. Always ensure your fractions are expressed with a common denominator before performing subtraction.

Simplifying Expressions

Simplifying expressions is all about making them easier to understand and work with. In algebra, this often involves combining like terms or reducing equations. When expressions have the same base, like in our exercise, you can use exponent rules to simplify:

• When you divide expressions with the same base, subtract the exponents of the terms. This is why \(y^{\frac{5}{7}} \div y^{\frac{3}{4}}\) simplifies to \(y^{\frac{5}{7} - \frac{3}{4}}\).
• By using common denominator techniques from fraction subtraction, this becomes \(y^{-\frac{1}{28}}\).
• Also consider simplifying any coefficients or constants, ensuring all like terms are combined.

Learning to simplify can at first be daunting but breaking it down into these steps can help make algebra feel less like a chore and more like a fun puzzle. Always look for opportunities to reduce terms, and use the exponent rules to your advantage.

Negative Exponents

Negative exponents often confuse students at first, but they simply represent the reciprocal of a positive exponent. Here’s what happens when you encounter a negative exponent, such as in our final result of \(y^{-\frac{1}{28}}\):

• Negative exponents indicate that the base is on the wrong side of a fraction bar. For example, \(a^{-n}\) is the same as \(\frac{1}{a^n}\).
• In simplified terms, \(y^{-\frac{1}{28}}\), means \(\frac{1}{y^{\frac{1}{28}}}\), transforming the expression into its positive exponent counterpart.
• This operation doesn’t alter the base itself, just how you represent it.

Understanding negative exponents as reciprocals clears up much confusion and allows you to easily flip an expression's position in a fraction. With practice, handling negative exponents becomes as intuitive as other algebraic rules.