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Chapter 5: Problem 71

Write each expression using a positive exponent. $$y^{-3}$$

Short Answer

Expert verified
\(\frac{1}{y^3}\) is the expression with a positive exponent.

Step by step solution

01

Understanding Negative Exponents

Negative exponents indicate that the base is on the wrong side of a fraction line. Therefore, if you have a negative exponent, you take the reciprocal of the base to make the exponent positive.
02

Apply the Rule for Negative Exponents

To convert the expression \(y^{-3}\) to have a positive exponent, take the reciprocal of \(y\). This changes the exponent from \(-3\) to \(+3\).
03

Write the Expression with a Positive Exponent

When you take the reciprocal, \(y^{-3}\) becomes \(\frac{1}{y^3}\). This expression now has a positive exponent.

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Key Concepts

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

Reciprocal
When we talk about reciprocal in mathematics, we are referring to a special kind of transformation of a number or expression.
In essence, the reciprocal of a number is 1 divided by that number.
For example, the reciprocal of a number like 5 is \( \frac{1}{5} \). It's like flipping the number upside down inside a fraction.

But why is this crucial, especially when dealing with negative exponents?
  • Negative exponents tell us to take the reciprocal of the base. This means, if you have an expression like \( y^{-3} \), the negative exponent tells you to take the reciprocal of \( y \).
  • By doing this, \( y^{-3} \) becomes \( \frac{1}{y^3} \), effectively turning the exponent positive.
Understanding how the reciprocal plays a role in transforming negative exponents into positive ones is key in algebra and various other areas of math.
Positive Exponents
Positive exponents indicate how many times you multiply a number by itself.
For instance, \( y^3 \) means that you multiply \( y \times y \times y \).
When dealing with expressions like \( y^{-3} \), converting to a positive exponent makes calculations straightforward.

Here's a simple breakdown:
  • Given a negative exponent, you initially have \( y^{-3} \).
  • Taking the reciprocal flips the expression, allowing the exponent to transform into a positive: \( \frac{1}{y^3} \).
  • Now with \( y^3 \) in the denominator, you can think of it as multiplying \( y \) three times.
This conversion not only simplifies mathematical operations but also aides in understanding the behavior of numbers under various operations.
Algebra
Algebra is a branch of mathematics where we use symbols (like \( x \) or \( y \)) to represent numbers and express mathematical concepts.
It's like a language for describing math, allowing us to work with numbers more generally and flexibly.

In the context of negative exponents and reciprocals:
  • Algebraic rules guide us on how to transform and simplify expressions, important when dealing with negative exponents.
  • When you come across \( y^{-3} \), using algebraic understanding, you know to apply the reciprocal to rewrite it with a positive exponent: \( \frac{1}{y^3} \).
  • Algebra helps in grasping the broader picture, seeing how each operation fits together and prepares you for more complex problems.
Mastering algebraic principles like these forms the foundation for success in advanced math topics and real-world problem-solving.

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Most popular questions from this chapter

Solve each equation. Check your solution. $$4.1 p=16.4$$

Use exponents to write the prime factorization of each number or monomial. \(42 a^{2} b\)

Write each decimal as a fraction or mixed number in simplest form. $$3.04$$

Estimate each product. $$1 \frac{2}{3} \cdot 4 \frac{1}{8}$$

Solve each equation. Check your solution. $$x-\frac{2}{5}=-\frac{8}{15}$$
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