/*! 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 39 Evaluate each exponential expres... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 39

Evaluate each exponential expression. $$ \left(\frac{3}{4}\right)^{4} $$

Short Answer

Expert verified
\( \frac{81}{256} \)

Step by step solution

01

Understand the Expression

Identify the base and the exponent in the expression. In \(\frac{3}{4}\right)^{4}\), the base is \( \frac{3}{4} \) and the exponent is 4.
02

Write the Expression in Expanded Form

Rewrite the expression by multiplying the base by itself as many times as indicated by the exponent. \(\frac{3}{4}\right)^{4}\) can be written as \( \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} \).
03

Multiply the Numerators

Multiply the numerators together: \( 3 \times 3 \times 3 \times 3 = 3^4 = 81 \).
04

Multiply the Denominators

Multiply the denominators together: \( 4 \times 4 \times 4 \times 4 = 4^4 = 256 \).
05

Form the Fraction

Combine the results from steps 3 and 4 into a single fraction: \(\frac{81}{256}\).

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.

base and exponent
To understand exponential expressions, it's crucial to know the roles of the base and the exponent. In the expression \(\frac{3}{4})^{4}\), the base is \(\frac{3}{4}\), which is the fraction being multiplied. The exponent is 4, which tells us how many times to multiply the base by itself.
  • The base can be any number, including fractions like in our example.
  • The exponent, also known as the 鈥榩ower,鈥� is always a non-negative integer in these cases.
This setup makes it simpler to manage and understand large multiplications through repeated steps.
writing in expanded form
Writing an exponential expression in expanded form means breaking it down into a series of multiplications. For \(\frac{3}{4})^{4}\), you'll write:
\(\frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4}\)
  • This helps to visualize the expression and makes it easier to carry out the multiplications step-by-step.
  • Expanded form reveals the structure of the problem, showing exactly how many times the base is multiplied by itself.
Taking this approach provides a clear path to simplifying complex exponential expressions.
multiplying fractions
When multiplying fractions, you multiply the numerators and denominators separately. For \(\frac{3}{4})^{4}\), this becomes:
\begin{align*} \text{Numerators: } & 3 \times 3 \times 3 \times 3 = 3^4 = 81 \ \text{Denominators: } & 4 \times 4 \times 4 \times 4 = 4^4 = 256 \ \frac{81}{256} \ \end{align*}
  • Remember to simplify the fractions if possible after you finish the multiplication.
  • In this case, \(\frac{81}{256}\) is already in its simplest form.
Breaking down the steps will make it much easier to handle and understand the problem, making the process less intimidating.

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

Evaluate each exponential expression. $$ (-4)^{5} $$

Use row operations to solve each system. $$ \begin{array}{r} {2 x-4 y=8} \\ {-3 x+6 y=5} \end{array} $$

Solve each system by substitution. If the system is inconsistent or has dependent equations, say so. See Examples 3鈥�5, 8, and 9. $$ \begin{aligned} &y=2 x\\\ &4 x-2 y=0 \end{aligned} $$

For each system, (a) solve by elimination or substitution and (b) use a graphing calculator to support your result. In part (b), be sure to solve each equation for y first. $$ \begin{array}{l} {6 x+y=5} \\ {-x+y=-9} \end{array} $$

Use row operations to solve each system. $$ \begin{aligned} &x-y=1\\\ &y-z=6\\\ &x+z=-1 \end{aligned} $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Logic and Functions

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.