Problem 55 Rewrite the radical expression i... [FREE SOLUTION]

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Elementary Algebra

Laura Bracken, Ed Miller

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 8: Problem 55

Rewrite the radical expression in exponential notation and simplify. $$\sqrt[4]{x^{2}}$$

Short Answer

Expert verified

The expression simplifies to $\sqrt{x}$.

Step by step solution

Rewrite the Radical as a Fractional Exponent

The expression $\sqrt[4]{x^2}$ can be rewritten using exponential notation by expressing the root as a fractional exponent. The fourth root of $x^2$ can be written as $x^{2/4}$.

Simplify the Fraction

Simplify the fractional exponent $2/4$ by dividing both the numerator and the denominator by their greatest common divisor. $\frac{2}{4} = \frac{1}{2}$. Thus, the expression $x^{2/4}$ simplifies to $x^{1/2}$.

Express as a Radical

The exponent $1/2$ indicates a square root. Therefore, $x^{1/2}$ can be written as $\sqrt{x}$.

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Key Concepts

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

Fractional Exponents

Fractional exponents are another way to represent roots and powers in a more compact form. For example, the nth root of a number is the same as raising that number to the power of 1/n.
This means that the fourth root of a number can be written as that number raised to the power of 1/4.
For instance, $\text{the fourth root of } x^2 $ can be written as $\text{x}^{2/4} $. By doing this, we turn a radical expression into something we can work with more easily when using algebraic rules for exponents.
Changing a root to a fractional exponent lets us apply exponent rules, such as multiplying and dividing exponents, which can simplify our calculations.

Simplifying Fractions

Simplifying fractions is a process where you reduce a fraction to its simplest form. This means making the top and bottom numbers (numerator and denominator) as small as possible.

Step 1: Find the greatest common divisor (GCD) of the numerator and denominator. The GCD is the largest number that divides both numbers without leaving a remainder.

For example, in the fraction $\frac{2}{4}$, the GCD of 2 and 4 is 2.

Step 2: Divide both the numerator and the denominator by their GCD. $\frac{2}{4} = \frac{2梅2}{4梅2} = \frac{1}{2}$

This gives you the simplest form of the fraction.
By simplifying the fraction $\frac{2}{4}$ to $\frac{1}{2}$, we make it easier to work with in our expressions, such as in our exercise when transforming $\text{x}^{2/4}$ into $\text{x}^{1/2}$.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. Represented by the radical symbol (鈭�), square roots are a foundational concept in algebra and many other areas of mathematics.
For example, $\text{the square root of 9 is 3 because 3 times 3 equals 9}$.
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91影视

Rewrite the radical expression in exponential notation and simplify. $$\sqrt[4]{x^{2}}$$

Short Answer

Step by step solution

Rewrite the Radical as a Fractional Exponent

Simplify the Fraction

Express as a Radical

Key Concepts

Fractional Exponents

Simplifying Fractions

Square Roots

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