Solved on Feb 11, 2024

Find the value of the expression $\frac{6^{\frac{5}{2}}}{6}$ .

STEP 1

Assumptions
1. We are given the expression $\frac{6^{\frac{5}{2}}}{6}$ .
2. We need to simplify the expression.

STEP 2

Recognize that the expression can be simplified by using the properties of exponents. Specifically, we can use the property that $\frac{a^m}{a^n} = a^{m-n}$ , where $a$ is the base and $m$ and $n$ are exponents.

STEP 3

Apply the property of exponents to the given expression.
$\frac{6^{\frac{5}{2}}}{6} = 6^{\frac{5}{2} - 1}$

STEP 4

Subtract the exponents, remembering that $6$ is the same as $6^1$ .
$6^{\frac{5}{2} - 1} = 6^{\frac{5}{2} - \frac{2}{2}}$

STEP 5

Perform the subtraction in the exponent.
$6^{\frac{5}{2} - \frac{2}{2}} = 6^{\frac{3}{2}}$

STEP 6

Recognize that $6^{\frac{3}{2}}$ can be rewritten as $(6^{\frac{1}{2}})^3$ since $(a^{\frac{1}{n}})^m = a^{\frac{m}{n}}$ .
$6^{\frac{3}{2}} = (6^{\frac{1}{2}})^3$

STEP 7

Calculate $6^{\frac{1}{2}}$ , which is the square root of $6$ .
$6^{\frac{1}{2}} = \sqrt{6}$

STEP 8

Raise the square root of $6$ to the power of $3$ .
$(\sqrt{6})^3 = (\sqrt{6})(\sqrt{6})(\sqrt{6})$

STEP 9

Multiply the square roots of $6$ together.
$(\sqrt{6})(\sqrt{6})(\sqrt{6}) = 6\sqrt{6}$
The simplified form of the expression $\frac{6^{\frac{5}{2}}}{6}$ is $6\sqrt{6}$ .

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Find the value of the expression $\frac{6^{\frac{5}{2}}}{6}$ .

STEP 1

Assumptions
1. We are given the expression $\frac{6^{\frac{5}{2}}}{6}$ .
2. We need to simplify the expression.

STEP 2

Recognize that the expression can be simplified by using the properties of exponents. Specifically, we can use the property that $\frac{a^m}{a^n} = a^{m-n}$ , where $a$ is the base and $m$ and $n$ are exponents.

STEP 3

Apply the property of exponents to the given expression.
$\frac{6^{\frac{5}{2}}}{6} = 6^{\frac{5}{2} - 1}$

STEP 4

Subtract the exponents, remembering that $6$ is the same as $6^1$ .
$6^{\frac{5}{2} - 1} = 6^{\frac{5}{2} - \frac{2}{2}}$

STEP 5

Perform the subtraction in the exponent.
$6^{\frac{5}{2} - \frac{2}{2}} = 6^{\frac{3}{2}}$

STEP 6

Recognize that $6^{\frac{3}{2}}$ can be rewritten as $(6^{\frac{1}{2}})^3$ since $(a^{\frac{1}{n}})^m = a^{\frac{m}{n}}$ .
$6^{\frac{3}{2}} = (6^{\frac{1}{2}})^3$

STEP 7

Calculate $6^{\frac{1}{2}}$ , which is the square root of $6$ .
$6^{\frac{1}{2}} = \sqrt{6}$

STEP 8

Raise the square root of $6$ to the power of $3$ .
$(\sqrt{6})^3 = (\sqrt{6})(\sqrt{6})(\sqrt{6})$

STEP 9

Multiply the square roots of $6$ together.
$(\sqrt{6})(\sqrt{6})(\sqrt{6}) = 6\sqrt{6}$
The simplified form of the expression $\frac{6^{\frac{5}{2}}}{6}$ is $6\sqrt{6}$ .

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Find the value of the expression 6526\frac{6^{\frac{5}{2}}}{6}6625​​.

STEP 1

Assumptions1. We are given the expression 6526\frac{6^{\frac{5}{2}}}{6}6625​​.2. We need to simplify the expression.

STEP 2

Recognize that the expression can be simplified by using the properties of exponents. Specifically, we can use the property that aman=am−n\frac{a^m}{a^n} = a^{m-n}anam​=am−n, where aaa is the base and mmm and nnn are exponents.

STEP 3

Apply the property of exponents to the given expression.6526=652−1\frac{6^{\frac{5}{2}}}{6} = 6^{\frac{5}{2} - 1}6625​​=625​−1

STEP 4

Subtract the exponents, remembering that 666 is the same as 616^161.652−1=652−226^{\frac{5}{2} - 1} = 6^{\frac{5}{2} - \frac{2}{2}}625​−1=625​−22​

STEP 5

Perform the subtraction in the exponent.652−22=6326^{\frac{5}{2} - \frac{2}{2}} = 6^{\frac{3}{2}}625​−22​=623​

STEP 6

Recognize that 6326^{\frac{3}{2}}623​ can be rewritten as (612)3(6^{\frac{1}{2}})^3(621​)3 since (a1n)m=amn(a^{\frac{1}{n}})^m = a^{\frac{m}{n}}(an1​)m=anm​.632=(612)36^{\frac{3}{2}} = (6^{\frac{1}{2}})^3623​=(621​)3

STEP 7

Calculate 6126^{\frac{1}{2}}621​, which is the square root of 666.612=66^{\frac{1}{2}} = \sqrt{6}621​=6​

STEP 8

Raise the square root of 666 to the power of 333.(6)3=(6)(6)(6)(\sqrt{6})^3 = (\sqrt{6})(\sqrt{6})(\sqrt{6})(6​)3=(6​)(6​)(6​)

STEP 9

Multiply the square roots of 666 together.(6)(6)(6)=66(\sqrt{6})(\sqrt{6})(\sqrt{6}) = 6\sqrt{6}(6​)(6​)(6​)=66​The simplified form of the expression 6526\frac{6^{\frac{5}{2}}}{6}6625​​ is 666\sqrt{6}66​.

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Find the value of the expression $\frac{6^{\frac{5}{2}}}{6}$ .

Assumptions
1. We are given the expression $\frac{6^{\frac{5}{2}}}{6}$ .
2. We need to simplify the expression.

Recognize that the expression can be simplified by using the properties of exponents. Specifically, we can use the property that $\frac{a^m}{a^n} = a^{m-n}$ , where $a$ is the base and $m$ and $n$ are exponents.

Apply the property of exponents to the given expression.
$\frac{6^{\frac{5}{2}}}{6} = 6^{\frac{5}{2} - 1}$

Subtract the exponents, remembering that $6$ is the same as $6^1$ .
$6^{\frac{5}{2} - 1} = 6^{\frac{5}{2} - \frac{2}{2}}$

Perform the subtraction in the exponent.
$6^{\frac{5}{2} - \frac{2}{2}} = 6^{\frac{3}{2}}$

Recognize that $6^{\frac{3}{2}}$ can be rewritten as $(6^{\frac{1}{2}})^3$ since $(a^{\frac{1}{n}})^m = a^{\frac{m}{n}}$ .
$6^{\frac{3}{2}} = (6^{\frac{1}{2}})^3$

Calculate $6^{\frac{1}{2}}$ , which is the square root of $6$ .
$6^{\frac{1}{2}} = \sqrt{6}$

Raise the square root of $6$ to the power of $3$ .
$(\sqrt{6})^3 = (\sqrt{6})(\sqrt{6})(\sqrt{6})$

Multiply the square roots of $6$ together.
$(\sqrt{6})(\sqrt{6})(\sqrt{6}) = 6\sqrt{6}$
The simplified form of the expression $\frac{6^{\frac{5}{2}}}{6}$ is $6\sqrt{6}$ .