Problem 4 - Elementary and Intermediate Algebra

Question

Fill in the blanks. We read \(\left(-64 a^{5}\right)^{4 / 5}\) as "the quantity of \(-64 a^{5}\), _____ to the four-fifths power."

Step-by-Step Solution

Answer

raised

Step 1: Understanding the Expression

The expression given is \((-64 a^{5})^{4/5}\). It involves a base, \(-64 a^{5}\), and an exponent, \(4/5\). The task is to read the expression as a phrase that represents the mathematical operation.

Step 2: Breaking Down the Expression

The base of the expression is \(-64 a^{5}\) and represents the quantity enclosed in parentheses. The exponent \(4/5\) signifies a fractional power, which means we are taking the 'four-fifths' power of \(-64 a^{5}\).

Step 3: Translating into Words

In mathematical phrases, 'to the' followed by a fraction indicates the power to which the base is raised. Therefore, \((base)^{4/5}\) is read as 'the base to the four-fifths power.'

Key Concepts

Understanding Exponents

Exponents are a way to express repeated multiplication of the same number or variable. If you see something like

• \( a^b \), it means \( a \) is multiplied by itself \( b \) times.

When dealing with exponents, you will often encounter fractional exponents. The key to understanding them is realizing that they represent roots and powers.
For instance,

• \( x^{1/n} \) is equivalent to saying "the \( n \)-th root of \( x \)".
• \( x^{m/n} \) means "take the \( n \)-th root of \( x \) and then raise the result to the \( m \)th power."

In the exercise, when you read

• \((-64 a^{5})^{4/5}\), think of it as the fifth root of \((-64 a^{5})\), and then raise your answer to the fourth power.

Algebraic Expressions Explained

An algebraic expression is a combination of numbers, variables, and operational symbols. Think of it like a phrase in math, constructed to represent a value.

Expressions can be as simple as

• \( 3x \), which means three times the value of \( x \),

or as complex as the one in our exercise

• \((-64 a^{5})^{4/5}\).

In this expression:

• - \(-64\) is a constant; it stays the same no matter what value other variables may take.
• \(a^{5}\) means \(a \times a \times a \times a \times a\).

Combining these into an expression, you can modify or interpret them based on the exponents and other operations used, just like how we interpret it using the fractional exponent in the exercise.

Decoding Mathematical Notation

Mathematical notation is like a universal language for expressing mathematical ideas in a concise and clear way. It helps communicate complex operations simply and efficiently.

For example,

• \((-64 a^{5})^{4/5}\) in our exercise uses parentheses to clarify which part of the expression the exponent applies to.

Using proper notation, mathematicians can avoid ambiguity and ensure that expressions are interpreted correctly.
In notation:

• The parentheses indicate that everything inside should be treated as a single unit.
• The exponent outside the parentheses tells you what to do with that unit. Here, it means taking the fourth power of the fifth root.

This way, everyone can understand exactly which operations to perform and their order, avoiding any confusion in interpretation.