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Chapter 10: Problem 36

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

Short Answer

Expert verified
\( \frac{1}{y^{1/6}} \)

Step by step solution

01

Identify the Negative Exponent

The given expression is \( y^{-1/6} \). The exponent is \(-1/6\), which is negative, indicating that this expression represents the reciprocal of \( y^{1/6} \).
02

Apply the Rule of Negative Exponents

Use the rule that \( a^{-n} = \frac{1}{a^n} \). This means \( y^{-1/6} = \frac{1}{y^{1/6}} \). This new expression has a positive exponent.
03

Verify Simplification Possibility

Check if the expression \( \frac{1}{y^{1/6}} \) can be simplified further. Since there are no like terms or operations needed on the current expression, it is already in its simplest form.

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Key Concepts

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

Understanding Positive Exponents
Exponents are a powerful tool in mathematics that help us express repeated multiplication in a compact form. When we talk about **positive exponents**, we're dealing with the multiplication of a base number by itself a certain number of times. For instance, the expression \( y^3 \) indicates that the base \( y \) is multiplied by itself three times (\( y \times y \times y \)). Each positive exponent tells us "how many times" to use the base in a multiplication operation. Unlike negative exponents, positive exponents are straightforward: they simply reflect repeated multiplication without altering the numerator or denominator in a fractional expression.
Reciprocal and Negative Exponents
A reciprocal helps us understand negative exponents better. **Negative exponents** are a way to express division instead of multiplication and are closely tied to the concept of reciprocal. A negative exponent flips the base number to its reciprocal. For example, the expression \( a^{-n} \) means \( \frac{1}{a^n} \). This shows that instead of multiplying, we're dividing by that power. The original problem \( y^{-1/6} \) turns into \( \frac{1}{y^{1/6}} \) by following this rule. This transformation is key in moving from negative to positive exponents, making it easier to work with various mathematical operations.
Simplification of Expressions
**Simplification** in mathematics means reducing an expression to its simplest form. This makes it easier to interpret or solve further. In the given problem, the expression \( \frac{1}{y^{1/6}} \) represents the simplified form of the original expression with a negative exponent. While simplifying, you aim to remove any complexity such as reducing fractions, combining like terms, or eliminating negative exponents. In this case, no further simplification is possible because \( \frac{1}{y^{1/6}} \) is already as simple as can be. Always remember: the goal of simplification is to make expressions easier to handle while ensuring they retain their mathematical equivalency.

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Most popular questions from this chapter

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