Problem 36 - Intermediate Algebra

Question

Write with positive exponents. Simplify if possible. $$ y^{-1 / 6} $$

Step-by-Step Solution

Answer

The expression with positive exponents is \( \frac{1}{y^{1/6}} \).

Step 1: Understanding the Negative Exponent

The expression we have is \( y^{-1/6} \). A negative exponent suggests that we take the reciprocal of the base raised to the positive of that exponent. This means \( y^{-1/6} \) can be rewritten as \( \frac{1}{y^{1/6}} \).

Step 2: Express with Positive Exponents

Now, we have the expression \( \frac{1}{y^{1/6}} \). Here, there are no negative exponents remaining, as the previous negative exponent has been converted to a positive exponent in the denominator.

Key Concepts

Negative Exponents

Negative exponents can initially seem confusing, but with a simple rule, they become easy to handle. Instead of multiplying the base number multiple times, a negative exponent tells us to take the reciprocal of the base number. When you encounter a negative exponent, just remember:

• A negative exponent means move the base to the opposite part of the fraction. If it's in the numerator, move it to the denominator, and vice versa.
• The exponent becomes positive during this process.

For example, with the expression \(x^{-3}\), you would express this as \(\frac{1}{x^3}\). Here, the negative sign has simply sent the base, \(x\), into the denominator as a positive exponent. Understanding this concept will help you simplify expressions more easily.

Positive Exponents

When an exponent is positive, it indicates how many times you multiply the base by itself. Positive exponents are straightforward:

• If you have \(a^b\), it means "multiply \(a\) by itself \(b\) times."
• No reciprocal is needed because the base remains where it starts: in the numerator.

For example, \(2^3\) equals \(2 \times 2 \times 2 = 8\). This simplicity is why expressing numbers with positive exponents often makes them easier to work with in calculations and keeps equations neater and more recognizable.

Reciprocal

The term "reciprocal" is key when dealing with exponents, especially negative ones. A reciprocal is essentially "flipping" the fraction you’re working with:

• For a whole number or variable, the reciprocal is \(\frac{1}{x}\).
• If you have a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).

This switch is often used to turn negative exponents into positive ones. By taking the reciprocal of the base with a negative exponent, you can easily rewrite it with a positive exponent.This is exactly what happens when converting \(y^{-1/6}\) into \(\frac{1}{y^{1/6}}\). Knowing how to quickly find and use reciprocals makes working with exponents less daunting.

Simplification of Expressions

Simplifying expressions is all about making them easier to read and work with. When you encounter expressions with negative exponents or fractions, simplification becomes crucial:

• Start by converting negative exponents using the reciprocal method, transforming them into positive exponents.
• Then see if the expression can be evaluated or reduced further. Sometimes, converting negative exponents is the main step needed.

For instance, simplifying \(y^{-1/6}\) involves just removing the negative exponent through reciprocal conversion. By moving that base to the denominator and changing the sign of the exponent, the expression is simplified and easier to understand. Simplifying expressions ensures calculations remain accurate and more legible.