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Chapter 10: Problem 36

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

Short Answer

Expert verified
The equivalent expression is \( a^{3/2} \).

Step by step solution

01

Understand the radical expression

The expression given is \( \sqrt{a^{3}} \). This is a radical form, where the square root (\sqrt{}) is applied to \( a^3 \).
02

Recall the properties of exponents

Recall that the square root of a number can be written as a fractional exponent. Specifically, \( \sqrt{a} \) is equivalent to \( a^{1/2} \).
03

Apply the properties to the given expression

To rewrite \( \sqrt{a^{3}} \) using exponents, apply the property that \( \sqrt{x} = x^{1/2} \). Thus, \( \sqrt{a^{3}} \) is rewritten as \( \left(a^{3}\right)^{1/2} \).
04

Simplify the expression

Simplify the expression \( \left(a^{3}\right)^{1/2} \) by multiplying the exponents. Using the property \( \left(x^{m}\right)^{n} = x^{mn} \), we get: \( \left(a^{3}\right)^{1/2} = a^{3/2} \).

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Key Concepts

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

radical expressions
Radical expressions are mathematical expressions that involve roots, such as square roots or cube roots. In this exercise, we are working with a square root expression, specifically \( \sqrt{a^{3}} \). The radical symbol \( \sqrt{} \) stands for the root of a number. When there is no small number written in the upper left corner of the radical sign, it is understood to be a square root. This means we want to find the number which, when multiplied by itself, gives the number under the root. For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \).
fractional exponents
Fractional exponents are a way to express roots using exponents. They provide an alternative to the radical notation. For example, instead of writing \( \sqrt{a} \), we can write \( a^{1/2} \). This is based on the general property: \( \sqrt[n]{a} = a^{1/n} \). In our exercise, instead of writing \( \sqrt{a^{3}} \), we can use fractional exponents and write it as \( \left(a^{3}\right)^{1/2} \). This property is very useful because it allows us to apply the rules of exponents to simplify and manipulate expressions more easily.
properties of exponents
Understanding the properties of exponents is critical when working with exponential notation and simplifying expressions. There are several key properties to remember:
    \bullet \ \( x^{m} \times x^{n} = x^{m+n} \) - When multiplying like bases, add the exponents.
    \bullet \ \( x^{m} \/ x^{n} = x^{m-n} \) - When dividing like bases, subtract the exponents.
    \bullet \ \( (x^{m})^{n} = x^{mn} \) - When raising a power to another power, multiply the exponents.
In our specific exercise, we use the property \( (a^{m})^{n} = a^{mn} \) to simplify \( (a^{3})^{1/2} \). By multiplying the exponents, we get: \( a^{3\times 1/2} = a^{3/2} \). This demonstrates how powerful these properties are in converting and simplifying expressions.

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Most popular questions from this chapter

Review simplifying rational expressions (Sections\(7.1-7.5)\) Perform the indicated operation and, if possible, simplify. $$ \frac{15 a^{2} x}{8 b} \cdot \frac{24 b^{2} x}{5 a}[7.2] $$

Perform the indicated operation and simplify. Assume that all variables represent positive real numbers. $$\frac{\sqrt[3]{(2 x+1)^{2}}}{\sqrt[5]{(2 x+1)^{2}}}$$

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