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Chapter 3: Problem 63

Evaluate each expression. See Sections 1.3 and 1.4 $$ \left(\frac{3}{5}\right)^{3} $$

Short Answer

Expert verified
The expression evaluates to \( \frac{27}{125} \).

Step by step solution

01

Understand the base and exponent

The expression \( \left(\frac{3}{5}\right)^{3} \) involves a base of \( \frac{3}{5} \) and an exponent of 3. This means we need to multiply \( \frac{3}{5} \) by itself three times.
02

Multiply the base by itself

Multiply \( \frac{3}{5} \) by itself: \( \frac{3}{5} \times \frac{3}{5} = \frac{3 \times 3}{5 \times 5} = \frac{9}{25} \).
03

Multiply by the base again

Take the result from Step 2 and multiply by \( \frac{3}{5} \) again: \( \frac{9}{25} \times \frac{3}{5} = \frac{9 \times 3}{25 \times 5} = \frac{27}{125} \).

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Key Concepts

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

Fractions
A fraction represents a part of a whole and is composed of two numbers - the numerator and the denominator.
The numerator is the top number, which indicates how many parts you have, while the denominator, the bottom number, shows how many equal parts the whole is divided into.
For example, in \( \frac{3}{5} \), 3 is the numerator and 5 is the denominator, indicating that the whole is divided into 5 parts and 3 parts are being considered.
Understanding fractions is crucial as they are used in various mathematical operations including addition, subtraction, multiplication, and division.
Multiplying Fractions
Multiplying fractions involves a straightforward process compared to addition and subtraction.
  • To multiply two fractions, multiply the numerators (top numbers) together and the denominators (bottom numbers) together.
  • For example, to multiply \( \frac{3}{5} \) by itself, you perform the operation \( \frac{3 \times 3}{5 \times 5} = \frac{9}{25} \).

It's important to simplify your answer if possible, but in this case, \( \frac{9}{25} \) does not need simplification.
Multiplying fractions is foundational for understanding more complex operations involving algebraic expressions.
Base and Exponent
In mathematics, the base and exponent are used to express repeated multiplication concisely.
The base is the number that gets multiplied, and the exponent tells how many times to multiply the base by itself.
  • For instance, in \( \left(\frac{3}{5}\right)^3 \), \( \frac{3}{5} \) is the base and 3 is the exponent.
  • This expression means \( \frac{3}{5} \times \frac{3}{5} \times \frac{3}{5} \).

This repeated multiplication results in \( \frac{27}{125} \). Utilizing exponents simplifies expressions and calculations, particularly in algebra and higher-level math.
Algebraic Expressions
An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols.
They provide a way to represent real-world problems and their solutions in mathematical form.
While the exercise given has a specific numeric expression, similar principles apply in algebraic contexts.
  • For example, if \( a = \frac{3}{5} \), then \( a^3 \) would be calculated the same as \( \left(\frac{3}{5}\right)^3 \).
  • Understanding how to manipulate fractions and exponents within these expressions is vital for solving and simplifying algebraic equations.

Mastering these concepts will aid in tackling more complex algebraic problems efficiently.

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Most popular questions from this chapter

The function \(f(x)=0.42 x+10.5,\) can be used to predict diamond production. For this function, \(x\) is the number of years after \(2000,\) and \(f(x)\) is the value (in billions of dollars) of the years diamond production. Use the function to predict diamond production in 2015 .

a. On a single screen, graph \(y=\frac{1}{2} x+1, y=x+1,\) and \(y=2 x+1 .\) Notice the change in slope for each graph. b. On a single screen, graph \(y=-\frac{1}{2} x+1, y=-x+1,\) and \(y=-2 x+1 .\) Notice the change in slope for each graph. c. Determine whether the following statement is true or false for slope \(m\) of a given line. As \(|m|\) becomes greater, the line becomes steeper.

Solve \- Chris-Craft manufactures boats out of Fiberglas and wood. Fiberglas hulls require 2 hours of work, whereas wood hulls require 4 hours of work. Employees work at most \(-40\) hours a week. The following inequalities model these restrictions, where \(x\) represents the number of Fiberglas hulls produced and \(y\) represents the number of wood hulls produced. $$ \left\\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 2 x+4 y & \leq 40 \end{aligned}\right. $$

The function \(f(x)=0.42 x+10.5,\) can be used to predict diamond production. For this function, \(x\) is the number of years after \(2000,\) and \(f(x)\) is the value (in billions of dollars) of the years diamond production. Use the function to predict diamond production in 2012 .

Find an equation of the perpendicular bisector of the line segment whose endpoints are given. \((-6,-3) ;(-8,-1)\)
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