Problem 43 Rewrite the radical expression i... [FREE SOLUTION]

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Elementary Algebra

Laura Bracken, Ed Miller

$Math Studyset 91影视 Explanations$ Math

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Chapter 8: Problem 43

Rewrite the radical expression in exponential notation. $$\sqrt{x^{3}}$$

Short Answer

Expert verified

x^{\frac{3}{2}}

Step by step solution

Identify the Radical

Observe that the expression includes a square root radical applied to the term inside it, which is $x^3$.

Convert the Radical to Exponential Notation

Rewrite the square root radical as an exponent of \frac{1}{2}\text{. Since we have $x^3$ under the square root, we will have \[ \left( x^3 \right)^{\frac{1}{2}} \].

Apply Exponent Rules

Use the Power of a Power rule to simplify: \[ \left( x^3 \right)^{\frac{1}{2}} = x^{3 \times \frac{1}{2}} = x^{\frac{3}{2}} \].

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Key Concepts

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

Exponential Notation

Exponential notation is a way to represent numbers, especially large and small ones, using powers of 10 or other bases. In the context of algebra and radicals, it simplifies expressions involving roots into exponential form. For instance, the square root of a number can be expressed exponentially. This simplifies calculations and allows us to use exponent rules.
In our exercise, we鈥檙e starting with the radical expression $\sqrt{x^3}$.By converting the square root into exponential notation, we rewrite it as follows: \[ (x^3)^{\frac{1}{2}} \]This notation makes it easier to apply rules for exponents during further simplifications.

Square Root

A square root is a special type of root that finds a number which, when multiplied by itself, gives the original number. The square root of $x$ is written as $ \sqrt{x}$.
In our example, we have a square root: $ \sqrt{x^3}$. This can be tricky because it involves a variable expression inside the square root.
To make it easier to handle, we convert the square root into a fractional exponent. For any number $a$, the square root is the same as raising it to the power of 1/2:

$ a = a^{1/2} $

This means we can rewrite our square root as an exponent: \[ \sqrt{x^3} = (x^3)^{\frac{1}{2}} \]

Power Rule

The power rule in exponents helps us simplify expressions where exponents are raised to other exponents. When you have an expression like $(a^m)^n$, you can multiply the exponents: \[ (a^m)^n = a^{m*n} \]Let's apply this rule to our example:\[ (x^3)^{\frac{1}{2}} \]We'll use the power rule here: Multiply 3 (the exponent from $x^3$) by $\frac{1}{2}$ (the exponent from the square root).So, it becomes: \[ x^{3 * \frac{1}{2}} = x^{\frac{3}{2}} \]By using the power rule, we have simplified the radical expression to an easier-to-handle exponential notation.

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91影视

Rewrite the radical expression in exponential notation. $$\sqrt{x^{3}}$$

Short Answer

Step by step solution

Identify the Radical

Convert the Radical to Exponential Notation

Apply Exponent Rules

Key Concepts

Exponential Notation

Square Root

Power Rule

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