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Rewrite in simplest rational exponent form [tex]\sqrt{x} \cdot \sqrt[4]{x}[/tex]. Show each step of your process.

Sagot :

Sure! Let's simplify the expression [tex]\(\sqrt{x} \cdot \sqrt[4]{x}\)[/tex] into its simplest rational exponent form step by step.

1. First, we rewrite [tex]\(\sqrt{x}\)[/tex] and [tex]\(\sqrt[4]{x}\)[/tex] using rational exponents.

- The square root of [tex]\(x\)[/tex] can be written as [tex]\(x\)[/tex] raised to the power of [tex]\(\frac{1}{2}\)[/tex]:
[tex]\[ \sqrt{x} = x^{\frac{1}{2}} \][/tex]

- The fourth root of [tex]\(x\)[/tex] can be written as [tex]\(x\)[/tex] raised to the power of [tex]\(\frac{1}{4}\)[/tex]:
[tex]\[ \sqrt[4]{x} = x^{\frac{1}{4}} \][/tex]

2. Next, we multiply these two terms together:
[tex]\[ x^{\frac{1}{2}} \cdot x^{\frac{1}{4}} \][/tex]

3. To multiply exponential terms with the same base, we add their exponents:
[tex]\[ x^{\frac{1}{2}} \cdot x^{\frac{1}{4}} = x^{\left(\frac{1}{2} + \frac{1}{4}\right)} \][/tex]

4. We now need to add the exponents [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex]:

- First, find a common denominator. The least common denominator of 2 and 4 is 4:
[tex]\[ \frac{1}{2} = \frac{2}{4} \][/tex]

- Now, add the fractions:
[tex]\[ \frac{2}{4} + \frac{1}{4} = \frac{3}{4} \][/tex]

5. Therefore, [tex]\(\frac{1}{2} + \frac{1}{4} = \frac{3}{4}\)[/tex]. We can now write the expression with the simplified exponent:

[tex]\[ x^{\left(\frac{3}{4}\right)} \][/tex]

Hence, the simplest rational exponent form of [tex]\(\sqrt{x} \cdot \sqrt[4]{x}\)[/tex] is:
[tex]\[ x^{\frac{3}{4}} \][/tex]