Question Find the equivalent expression for $\sqrt[5]{x^{41}}$ when $x>0$ .

Studdy Solution

STEP 1

Assumptions
1. The variable $x$ is a positive real number ( $x > 0$ ).
2. We are dealing with real numbers and their properties.
3. We need to simplify the expression $\sqrt[5]{x^{41}}$ .

STEP 2

Understand the properties of exponents and roots. The fifth root of $x$ raised to a power can be expressed as $x$ raised to the power divided by 5.
$\sqrt[5]{x^n} = x^{\frac{n}{5}}$

STEP 3

Apply the property of exponents to the given expression $\sqrt[5]{x^{41}}$ .
$\sqrt[5]{x^{41}} = x^{\frac{41}{5}}$

STEP 4

Divide the exponent 41 by 5 to find the equivalent expression.
$x^{\frac{41}{5}} = x^{8 + \frac{1}{5}}$

STEP 5

Recognize that $x^{8 + \frac{1}{5}}$ can be separated into two parts using the property of exponents that states $x^{a+b} = x^a \cdot x^b$ .
$x^{8 + \frac{1}{5}} = x^8 \cdot x^{\frac{1}{5}}$

STEP 6

Understand that $x^{\frac{1}{5}}$ is the fifth root of $x$ .
$x^{\frac{1}{5}} = \sqrt[5]{x}$

STEP 7

Combine the results from STEP_5 and STEP_6 to express the simplified form of the original expression.
$\sqrt[5]{x^{41}} = x^8 \cdot \sqrt[5]{x}$
The expression $\sqrt[5]{x^{41}}$ is equivalent to $x^8 \cdot \sqrt[5]{x}$ .

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Question Find the equivalent expression for $\sqrt[5]{x^{41}}$ when $x>0$ .

Studdy Solution

STEP 1

Assumptions
1. The variable $x$ is a positive real number ( $x > 0$ ).
2. We are dealing with real numbers and their properties.
3. We need to simplify the expression $\sqrt[5]{x^{41}}$ .

STEP 2

Understand the properties of exponents and roots. The fifth root of $x$ raised to a power can be expressed as $x$ raised to the power divided by 5.
$\sqrt[5]{x^n} = x^{\frac{n}{5}}$

STEP 3

Apply the property of exponents to the given expression $\sqrt[5]{x^{41}}$ .
$\sqrt[5]{x^{41}} = x^{\frac{41}{5}}$

STEP 4

Divide the exponent 41 by 5 to find the equivalent expression.
$x^{\frac{41}{5}} = x^{8 + \frac{1}{5}}$

STEP 5

Recognize that $x^{8 + \frac{1}{5}}$ can be separated into two parts using the property of exponents that states $x^{a+b} = x^a \cdot x^b$ .
$x^{8 + \frac{1}{5}} = x^8 \cdot x^{\frac{1}{5}}$

STEP 6

Understand that $x^{\frac{1}{5}}$ is the fifth root of $x$ .
$x^{\frac{1}{5}} = \sqrt[5]{x}$

STEP 7

Combine the results from STEP_5 and STEP_6 to express the simplified form of the original expression.
$\sqrt[5]{x^{41}} = x^8 \cdot \sqrt[5]{x}$
The expression $\sqrt[5]{x^{41}}$ is equivalent to $x^8 \cdot \sqrt[5]{x}$ .

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Question Find the equivalent expression for x415\sqrt[5]{x^{41}}5x41​ when x>0x>0x>0.

Studdy Solution

STEP 1

Assumptions1. The variable xxx is a positive real number (x>0x > 0x>0).2. We are dealing with real numbers and their properties.3. We need to simplify the expression x415\sqrt[5]{x^{41}}5x41​.

STEP 2

Understand the properties of exponents and roots. The fifth root of xxx raised to a power can be expressed as xxx raised to the power divided by 5.xn5=xn5\sqrt[5]{x^n} = x^{\frac{n}{5}}5xn​=x5n​

STEP 3

Apply the property of exponents to the given expression x415\sqrt[5]{x^{41}}5x41​.x415=x415\sqrt[5]{x^{41}} = x^{\frac{41}{5}}5x41​=x541​

STEP 4

Divide the exponent 41 by 5 to find the equivalent expression.x415=x8+15x^{\frac{41}{5}} = x^{8 + \frac{1}{5}}x541​=x8+51​

STEP 5

Recognize that x8+15x^{8 + \frac{1}{5}}x8+51​ can be separated into two parts using the property of exponents that states xa+b=xa⋅xbx^{a+b} = x^a \cdot x^bxa+b=xa⋅xb.x8+15=x8⋅x15x^{8 + \frac{1}{5}} = x^8 \cdot x^{\frac{1}{5}}x8+51​=x8⋅x51​

STEP 6

Understand that x15x^{\frac{1}{5}}x51​ is the fifth root of xxx.x15=x5x^{\frac{1}{5}} = \sqrt[5]{x}x51​=5x​

STEP 7

Combine the results from STEP_5 and STEP_6 to express the simplified form of the original expression.x415=x8⋅x5\sqrt[5]{x^{41}} = x^8 \cdot \sqrt[5]{x}5x41​=x8⋅5x​The expression x415\sqrt[5]{x^{41}}5x41​ is equivalent to x8⋅x5x^8 \cdot \sqrt[5]{x}x8⋅5x​.

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Question Find the equivalent expression for $\sqrt[5]{x^{41}}$ when $x>0$ .

Assumptions
1. The variable $x$ is a positive real number ( $x > 0$ ).
2. We are dealing with real numbers and their properties.
3. We need to simplify the expression $\sqrt[5]{x^{41}}$ .

Understand the properties of exponents and roots. The fifth root of $x$ raised to a power can be expressed as $x$ raised to the power divided by 5.
$\sqrt[5]{x^n} = x^{\frac{n}{5}}$

Apply the property of exponents to the given expression $\sqrt[5]{x^{41}}$ .
$\sqrt[5]{x^{41}} = x^{\frac{41}{5}}$

Divide the exponent 41 by 5 to find the equivalent expression.
$x^{\frac{41}{5}} = x^{8 + \frac{1}{5}}$

Recognize that $x^{8 + \frac{1}{5}}$ can be separated into two parts using the property of exponents that states $x^{a+b} = x^a \cdot x^b$ .
$x^{8 + \frac{1}{5}} = x^8 \cdot x^{\frac{1}{5}}$

Understand that $x^{\frac{1}{5}}$ is the fifth root of $x$ .
$x^{\frac{1}{5}} = \sqrt[5]{x}$

Combine the results from STEP_5 and STEP_6 to express the simplified form of the original expression.
$\sqrt[5]{x^{41}} = x^8 \cdot \sqrt[5]{x}$
The expression $\sqrt[5]{x^{41}}$ is equivalent to $x^8 \cdot \sqrt[5]{x}$ .