Method

Simplifying Radical Expressions

Expressions with radicals can be written using rational exponents. Then, they can be simplified using the properties of exponents. Consider the following expression.

\frac{4 x ^{3} \cdot 3 x ^{4}}{6 x ^{7}}

Rewrite terms into

n a^{n}

When a term inside a radical has a power greater than the index of the radical, it can be rewritten into

n a^{n}

-form. In the example, there are two such terms,

3 x^{4}

and

6 x^{7} .

4 > 3 and 7 > 6

First, the Product of Powers Property can be used to rewrite the terms under these radicals.

\frac{4 x ^{3} \cdot 3 x ^{4}}{6 x ^{7}}

Rewrite

Rewrite $4$ as $3 + 1$

\frac{4 x ^{3} \cdot 3 x ^{3 + 1}}{6 x ^{7}}

Rewrite

Rewrite $7$ as $6 + 1$

\frac{4 x ^{3} \cdot 3 x ^{3 + 1}}{6 x ^{6 + 1}}

SumInExponent

$a^{m + n} = a^{m} \cdot a^{n}$

\frac{4 x ^{3} \cdot 3 x ^{3} \cdot x ^{1}}{6 x ^{6} \cdot x ^{1}}

The products under the radicals can now be rewritten using the Product Property of Radicals.

\frac{4 x ^{3} \cdot 3 x ^{3} \cdot x ^{1}}{6 x ^{6} \cdot x ^{1}}

RootProd

$3 a \cdot b = 3 a \cdot 3 b$

\frac{4 x ^{3} \cdot 3 x ^{3} \cdot 3 x ^{1}}{6 x ^{6} \cdot x ^{1}}

RootProd

$6 a \cdot b = 6 a \cdot 6 b$

\frac{4 x ^{3} \cdot 3 x ^{3} \cdot 3 x ^{1}}{6 x ^{6} \cdot 6 x ^{1}}

ExponentOne

$a^{1} = a$

\frac{4 x ^{3} \cdot 3 x ^{3} \cdot 3 x}{6 x ^{6} \cdot 6 x}

Simplify

n a^{n}

-terms

The

n a^{n}

-terms can now be simplified as follows.

n a^{n} = {∣ a ∣ if n is odd ∣ a ∣ if n is even

In the expression there are two terms that can be simplified using this rule.

\frac{4 x ^{3} \cdot 3 x ^{3} \cdot 3 x}{6 x ^{6} \cdot 6 x}

RootPowToNumber

$3 a^{3} = a$

\frac{4 x ^{3} \cdot x \cdot 3 x}{6 x ^{6} \cdot 6 x}

$6 a^{6} = ∣ a ∣$

\frac{4 x ^{3} \cdot x \cdot 3 x}{∣ x ∣ \cdot 6 x}

Express the radicals using rational exponents

Next, rewrite the radicals using rational exponents. The example can be rewritten as follows.

\frac{4 x ^{3} \cdot x \cdot 3 x}{∣ x ∣ \cdot 6 x} \Leftrightarrow \frac{x ^{\frac{3}{4}} \cdot x \cdot x ^{\frac{1}{3}}}{∣ x ∣ \cdot x ^{\frac{1}{6}}}

Simplify using the laws of exponents

When the expression is written using rational exponents, it can be simplified. When two terms with the same base are multiplied, the exponents are added according to the Product of Powers Property.

\frac{x ^{\frac{3}{4}} \cdot x \cdot x ^{\frac{1}{3}}}{∣ x ∣ \cdot x ^{\frac{1}{6}}}

$a = a^{1}$

\frac{x ^{\frac{3}{4}} \cdot x ^{1} \cdot x ^{\frac{1}{3}}}{∣ x ∣ \cdot x ^{\frac{1}{6}}}

MultPow

$a^{m} \cdot a^{n} = a^{m + n}$

\frac{x ^{\frac{3}{4} + 1 + \frac{1}{3}}}{∣ x ∣ \cdot x ^{\frac{1}{6}}}

To simplify the exponent further, which requires adding and subtracting fractions, the denominators must be made the same. Here, the least common denominator is

12 .

\frac{x ^{\frac{3}{4} + 1 + \frac{1}{3}}}{∣ x ∣ \cdot x ^{\frac{1}{6}}}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 3}{b \cdot 3}$

\frac{x ^{\frac{9}{1 2} + 1 + \frac{1}{3}}}{∣ x ∣ \cdot x ^{\frac{1}{6}}}

OneToFrac

Rewrite $1$ as $\frac{1 2}{1 2}$

\frac{x ^{\frac{9}{1 2} + \frac{1 2}{1 2} + \frac{1}{3}}}{∣ x ∣ \cdot x ^{\frac{1}{6}}}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 4}{b \cdot 4}$

\frac{x ^{\frac{9}{1 2} + \frac{1 2}{1 2} + \frac{4}{1 2}}}{∣ x ∣ \cdot x ^{\frac{1}{6}}}

AddFrac

Add fractions

\frac{x ^{\frac{2 5}{1 2}}}{∣ x ∣ \cdot x ^{\frac{1}{6}}}

Since the expressions in the numerator and the denominator have the same bases,

x,

they can be simplified. First, by using the Quotient of Powers Property, the expression is written as one term. To simplify the exponent, the denominators must then be made the same.

\frac{x ^{\frac{2 5}{1 2}}}{∣ x ∣ \cdot x ^{\frac{1}{6}}}

DivPow

$\frac{a ^{m}}{a ^{n}} = a^{m - n}$

\frac{x ^{\frac{2 5}{1 2} - \frac{1}{6}}}{∣ x ∣}

ExpandFrac

$\frac{a}{b} = \frac{a \cdot 2}{b \cdot 2}$

\frac{x ^{\frac{1 3}{1 2} - \frac{2}{1 2}}}{∣ x ∣}

SubFrac

Subtract fractions

\frac{x ^{\frac{1 1}{1 2}}}{∣ x ∣}

Rewrite expression using radicals

When the expression has been simplified completely, it can be rewritten using radicals.

\frac{x ^{\frac{1 1}{1 2}}}{∣ x ∣} = \frac{1 2 x ^{11}}{∣ x ∣}

This is the simplified expression.

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