/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 Simplify each expression. \(\l... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 16

Simplify each expression. \(\left(\frac{3}{8}\right)^{0}\)

Short Answer

Expert verified
\(\left(\frac{3}{8}\right)^{0} = 1\)

Step by step solution

01

Identify the Power of Zero Rule

The expression given is \(\left(\frac{3}{8}\right)^{0}\). Recall the rule that any non-zero base raised to the power of zero equals 1. This rule applies because multiplying a number by itself zero times (essentially, not multiplying it at all) results in 1.
02

Apply the Power of Zero Rule

Using the rule from Step 1, simplify \(\left(\frac{3}{8}\right)^{0}\) to 1. Since the base \(\frac{3}{8}\) is not zero, the power of zero results in the value of 1.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Power of Zero Rule
The Power of Zero Rule in mathematics states that any non-zero number raised to the power of zero is equal to 1. This might seem a bit strange at first, but it makes sense when you think about it in terms of exponents. An exponent signifies how many times to multiply the base by itself. So, when the exponent is zero, you're essentially not multiplying the number at all. That's why it defaults to 1.
  • Non-zero base to power zero: Always equals 1.
  • Zero base: The case where the base itself is zero can be tricky. Remember, the rule applies to non-zero numbers.

In our exercise, \left(\frac{3}{8}\right)^{0}\ is such a case. The base is \(\frac{3}{8}\), which is a non-zero fraction. Applying the Power of Zero Rule, the expression simplifies directly to 1.
Exponentiation
Exponentiation is a mathematical operation involving two numbers: the base and the exponent. The base is the number being multiplied, and the exponent tells us how many times the base multiplies itself. For instance, in the expression \(a^n\), \(a\) is the base and \(n\) is the exponent.
  • If \(n = 2\), then \(a^n = a \times a\).
  • If \(n = 3\), then \(a^n = a \times a \times a\).

For an exponent of zero, though, things change slightly. As we learned with the Power of Zero Rule, \(a^0 = 1\) for any \(a eq 0\). This is an essential tool for simplifying expressions in algebra and beyond.
Fractional Base
A fractional base means that the base of your expression is a fraction. This doesn't change the rules of exponents, but it might look a bit different. In our case, we have \(\left(\frac{3}{8}\right)^{0}\).
  • Base as a fraction: When the base is a fraction, treat it like any other number in terms of the power rules.
  • Fractional base to zero power: Just like with whole numbers, any non-zero fraction raised to the power of zero is always 1.

Even though fractions can sometimes feel more complex, they follow the same exponent rules. So in the problem provided, \(\left(\frac{3}{8}\right)^{0} = 1\), which highlights the elegance of the Power of Zero Rule across different types of numbers.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use the product rule for exponents to simplify each expression. Write the results using exponents. $$ b b^{2} b^{3} $$

Write number in scientific notation. \(23,000\)

Use the power of a product rule for exponents to simplify each expression. See Example \(9 .\) \(\left(-2 r^{2} s^{3}\right)^{3}\)

Classify each polynomial as a monomial, a binomial, a trinomial, or none of these. See Example \(1 .\) $$ x^{3} $$

Write number in scientific notation. \(1,700,000\)
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.