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Chapter 10: Problem 35

Write an equivalent expression using exponential notation. $$ \sqrt{x^{3}} $$

Short Answer

Expert verified
The equivalent expression using exponential notation is \[ x^{3/2} \].

Step by step solution

01

Understand the Problem

The goal is to write \(\textbackslash sqrt{x^{3}}\) using exponential notation.
02

Recall the Exponential Rule

The square root of any number or variable \( a \) can be written as an exponent: \[ \textbackslash sqrt{a} = a^{1/2} \]
03

Apply the Exponential Rule

Apply the rule to rewrite \(\textbackslash sqrt{x^{3}}\) as an exponent: \[ \textbackslash sqrt{x^{3}} = (x^{3})^{1/2} \]
04

Multiply the Exponents

Use the rule of exponents that states \( (a^m)^n = a^{m \cdot n} \). So, \[ (x^{3})^{1/2} = x^{3 \cdot 1/2} \]
05

Simplify the Expression

Multiply the exponents: \[ 3 \cdot 1/2 = 3/2 \]. Thus, \[ (x^{3})^{1/2} = x^{3/2} \]

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Key Concepts

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

Square Root Conversion
Understanding square root conversions is crucial when dealing with exponential notation. The square root of a number or variable can be expressed as a fractional exponent. For example, the square root of a variable \( x \) is written as \( x^{ 1 / 2 } \). This means that taking the square root is the same as raising to the power of \( 1 / 2 \).

This rule is essential, especially when simplifying expressions involving roots. Instead of working with square roots directly, converting them to fractional exponents can make the process more straightforward. For example, if you have \( \sqrt{ x^3 } \), you can rewrite it as \( ( x^3 )^{ 1 / 2 } \).
Exponent Rules
Exponent rules help simplify and solve expressions involving powers. One key rule is the power of a power rule, which states that \( ( a^ m )^ n = a^{ m } \).

For example, with \( ( x^3 )^{ 1 / 2 } \), you need to multiply the exponents: \( 3 1 / 2 \). This results in \( x^{ 3 / 2 } \). Note that multiplying the exponents simplifies the operation and makes the expression easier to manage.

Additionally, understanding how to multiply and add exponents is critical. Multiplying exponents can be broken down into these straightforward steps:
  • Identify the base of the exponent.
  • Multiply the numerical values of the exponents.
  • Rewrite the expression with the new exponent.
Applying such rules can simplify complex algebraic expressions significantly.
Simplifying Expressions
Simplifying expressions often involves reducing them to their simplest form. This can mean converting roots into fractional exponents or combining like terms. In situations where you're given \( \sqrt{ x^3 } \), converting this into \( ( x^3 )^{ 1 / 2 } \) and subsequently applying exponent rules, results in a simpler form: \( x^{ 3 / 2 } \).

When simplifying expressions, follow these steps:
  • Convert roots to fractional exponents.
  • Apply exponent rules to simplify powers of powers.
  • Combine like terms where possible.


This process not only makes solving equations easier but also ensures the solutions are presented in their most streamlined form, making them easier to understand and work with.

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Most popular questions from this chapter

f(x) and g(x) are as given. Find \((f \cdot g)(x) \cdot\) Assume that all variables represent non negativereal numbers. $$f(x)=x+\sqrt{7}, g(x)=x-\sqrt{7} $$

Each side of a regular octagon has length \(s .\) Find a formula for the distance \(d\) between the parallel sides of the octagon.

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