IDNLearn.com offers a collaborative platform for sharing and gaining knowledge. Find the information you need quickly and easily with our reliable and thorough Q&A platform.
Sagot :
Certainly! Let's simplify the expression [tex]\(\sqrt{x} \cdot \sqrt[4]{x}\)[/tex] and rewrite it in its simplest rational exponent form by following a detailed, step-by-step process.
### Step 1: Rewrite the radicals using rational exponents
First, we need to express the given radicals as exponents:
- The square root of [tex]\(x\)[/tex], denoted [tex]\(\sqrt{x}\)[/tex], can be rewritten as [tex]\(x^{\frac{1}{2}}\)[/tex].
- The fourth root of [tex]\(x\)[/tex], denoted [tex]\(\sqrt[4]{x}\)[/tex], can be rewritten as [tex]\(x^{\frac{1}{4}}\)[/tex].
So, the expression becomes:
[tex]\[ x^{\frac{1}{2}} \cdot x^{\frac{1}{4}} \][/tex]
### Step 2: Use the property of exponents to combine the terms
We use the property of exponents that states:
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
Applying this property to our expression, we get:
[tex]\[ x^{\frac{1}{2}} \cdot x^{\frac{1}{4}} = x^{\left(\frac{1}{2} + \frac{1}{4}\right)} \][/tex]
### Step 3: Add the exponents together
Next, we perform the addition of the exponents [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex]:
- [tex]\(\frac{1}{2}\)[/tex] is equivalent to [tex]\(0.5\)[/tex]
- [tex]\(\frac{1}{4}\)[/tex] is equivalent to [tex]\(0.25\)[/tex]
When we add these together:
[tex]\[ \frac{1}{2} + \frac{1}{4} = 0.5 + 0.25 = 0.75 \][/tex]
So, the combined exponent is [tex]\(0.75\)[/tex].
### Step 4: Rewrite the expression with the simplified exponent
After adding the exponents, our expression becomes:
[tex]\[ x^{0.75} \][/tex]
### Step 5: Convert the decimal exponent back to a fraction (optional)
Though [tex]\(0.75\)[/tex] is already a simplified form, it is often useful to convert it back to a fraction for clarity. The fraction equivalent of [tex]\(0.75\)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
Thus, the expression can also be written as:
[tex]\[ x^{\frac{3}{4}} \][/tex]
### Conclusion
The expression [tex]\(\sqrt{x} \cdot \sqrt[4]{x}\)[/tex] simplified in its simplest rational exponent form is:
[tex]\[ x^{0.75} \quad \text{or} \quad x^{\frac{3}{4}} \][/tex]
Both forms are correct and represent the same simplified expression.
### Step 1: Rewrite the radicals using rational exponents
First, we need to express the given radicals as exponents:
- The square root of [tex]\(x\)[/tex], denoted [tex]\(\sqrt{x}\)[/tex], can be rewritten as [tex]\(x^{\frac{1}{2}}\)[/tex].
- The fourth root of [tex]\(x\)[/tex], denoted [tex]\(\sqrt[4]{x}\)[/tex], can be rewritten as [tex]\(x^{\frac{1}{4}}\)[/tex].
So, the expression becomes:
[tex]\[ x^{\frac{1}{2}} \cdot x^{\frac{1}{4}} \][/tex]
### Step 2: Use the property of exponents to combine the terms
We use the property of exponents that states:
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
Applying this property to our expression, we get:
[tex]\[ x^{\frac{1}{2}} \cdot x^{\frac{1}{4}} = x^{\left(\frac{1}{2} + \frac{1}{4}\right)} \][/tex]
### Step 3: Add the exponents together
Next, we perform the addition of the exponents [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{1}{4}\)[/tex]:
- [tex]\(\frac{1}{2}\)[/tex] is equivalent to [tex]\(0.5\)[/tex]
- [tex]\(\frac{1}{4}\)[/tex] is equivalent to [tex]\(0.25\)[/tex]
When we add these together:
[tex]\[ \frac{1}{2} + \frac{1}{4} = 0.5 + 0.25 = 0.75 \][/tex]
So, the combined exponent is [tex]\(0.75\)[/tex].
### Step 4: Rewrite the expression with the simplified exponent
After adding the exponents, our expression becomes:
[tex]\[ x^{0.75} \][/tex]
### Step 5: Convert the decimal exponent back to a fraction (optional)
Though [tex]\(0.75\)[/tex] is already a simplified form, it is often useful to convert it back to a fraction for clarity. The fraction equivalent of [tex]\(0.75\)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
Thus, the expression can also be written as:
[tex]\[ x^{\frac{3}{4}} \][/tex]
### Conclusion
The expression [tex]\(\sqrt{x} \cdot \sqrt[4]{x}\)[/tex] simplified in its simplest rational exponent form is:
[tex]\[ x^{0.75} \quad \text{or} \quad x^{\frac{3}{4}} \][/tex]
Both forms are correct and represent the same simplified expression.
Thank you for participating in our discussion. We value every contribution. Keep sharing knowledge and helping others find the answers they need. Let's create a dynamic and informative learning environment together. Find precise solutions at IDNLearn.com. Thank you for trusting us with your queries, and we hope to see you again.
