Question Find the most simplified form of $(x^{1/3})(x^{1/6})$ : $\sqrt{x}$ , $1/x^9$ , $x^{1/9}$ , $x^{1/18}$

Studdy Solution

STEP 1

Assumptions
1. We are given the expression $\left(x^{\frac{1}{3}}\right)\left(x^{\frac{1}{6}}\right)$ .
2. We need to simplify the expression to its most simplified form.

STEP 2

To simplify the expression, we will use the property of exponents that states when multiplying two exponents with the same base, we add the exponents.
$x^a \cdot x^b = x^{a+b}$

STEP 3

Apply the property of exponents to the given expression.
$\left(x^{\frac{1}{3}}\right)\left(x^{\frac{1}{6}}\right) = x^{\frac{1}{3} + \frac{1}{6}}$

STEP 4

Find a common denominator to add the fractions $\frac{1}{3}$ and $\frac{1}{6}$ .
The common denominator for 3 and 6 is 6.

STEP 5

Express $\frac{1}{3}$ with the common denominator 6.
$\frac{1}{3} = \frac{2}{6}$

STEP 6

Now, add the exponents with the common denominator.
$\frac{2}{6} + \frac{1}{6} = \frac{3}{6}$

STEP 7

Simplify the fraction $\frac{3}{6}$ .
$\frac{3}{6} = \frac{1}{2}$

STEP 8

Now, we can write the simplified exponent with the base $x$ .
$x^{\frac{3}{6}} = x^{\frac{1}{2}}$

STEP 9

The expression $x^{\frac{1}{2}}$ is equivalent to the square root of $x$ .
$x^{\frac{1}{2}} = \sqrt{x}$
The most simplified form of the expression $\left(x^{\frac{1}{3}}\right)\left(x^{\frac{1}{6}}\right)$ is $\sqrt{x}$ .

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Question Find the most simplified form of $(x^{1/3})(x^{1/6})$ : $\sqrt{x}$ , $1/x^9$ , $x^{1/9}$ , $x^{1/18}$

Studdy Solution

STEP 1

Assumptions
1. We are given the expression $\left(x^{\frac{1}{3}}\right)\left(x^{\frac{1}{6}}\right)$ .
2. We need to simplify the expression to its most simplified form.

STEP 2

To simplify the expression, we will use the property of exponents that states when multiplying two exponents with the same base, we add the exponents.
$x^a \cdot x^b = x^{a+b}$

STEP 3

Apply the property of exponents to the given expression.
$\left(x^{\frac{1}{3}}\right)\left(x^{\frac{1}{6}}\right) = x^{\frac{1}{3} + \frac{1}{6}}$

STEP 4

Find a common denominator to add the fractions $\frac{1}{3}$ and $\frac{1}{6}$ .
The common denominator for 3 and 6 is 6.

STEP 5

Express $\frac{1}{3}$ with the common denominator 6.
$\frac{1}{3} = \frac{2}{6}$

STEP 6

Now, add the exponents with the common denominator.
$\frac{2}{6} + \frac{1}{6} = \frac{3}{6}$

STEP 7

Simplify the fraction $\frac{3}{6}$ .
$\frac{3}{6} = \frac{1}{2}$

STEP 8

Now, we can write the simplified exponent with the base $x$ .
$x^{\frac{3}{6}} = x^{\frac{1}{2}}$

STEP 9

The expression $x^{\frac{1}{2}}$ is equivalent to the square root of $x$ .
$x^{\frac{1}{2}} = \sqrt{x}$
The most simplified form of the expression $\left(x^{\frac{1}{3}}\right)\left(x^{\frac{1}{6}}\right)$ is $\sqrt{x}$ .

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Question Find the most simplified form of (x1/3)(x1/6)(x^{1/3})(x^{1/6})(x1/3)(x1/6): x\sqrt{x}x​, 1/x91/x^91/x9, x1/9x^{1/9}x1/9, x1/18x^{1/18}x1/18

Studdy Solution

STEP 1

Assumptions1. We are given the expression (x13)(x16)\left(x^{\frac{1}{3}}\right)\left(x^{\frac{1}{6}}\right)(x31​)(x61​).2. We need to simplify the expression to its most simplified form.

STEP 2

To simplify the expression, we will use the property of exponents that states when multiplying two exponents with the same base, we add the exponents.xa⋅xb=xa+bx^a \cdot x^b = x^{a+b}xa⋅xb=xa+b

STEP 3

Apply the property of exponents to the given expression.(x13)(x16)=x13+16\left(x^{\frac{1}{3}}\right)\left(x^{\frac{1}{6}}\right) = x^{\frac{1}{3} + \frac{1}{6}}(x31​)(x61​)=x31​+61​

STEP 4

Find a common denominator to add the fractions 13\frac{1}{3}31​ and 16\frac{1}{6}61​.The common denominator for 3 and 6 is 6.

STEP 5

Express 13\frac{1}{3}31​ with the common denominator 6.13=26\frac{1}{3} = \frac{2}{6}31​=62​

STEP 6

Now, add the exponents with the common denominator.26+16=36\frac{2}{6} + \frac{1}{6} = \frac{3}{6}62​+61​=63​

STEP 7

Simplify the fraction 36\frac{3}{6}63​.36=12\frac{3}{6} = \frac{1}{2}63​=21​

STEP 8

Now, we can write the simplified exponent with the base xxx.x36=x12x^{\frac{3}{6}} = x^{\frac{1}{2}}x63​=x21​

STEP 9

The expression x12x^{\frac{1}{2}}x21​ is equivalent to the square root of xxx.x12=xx^{\frac{1}{2}} = \sqrt{x}x21​=x​The most simplified form of the expression (x13)(x16)\left(x^{\frac{1}{3}}\right)\left(x^{\frac{1}{6}}\right)(x31​)(x61​) is x\sqrt{x}x​.

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Question Find the most simplified form of $(x^{1/3})(x^{1/6})$ : $\sqrt{x}$ , $1/x^9$ , $x^{1/9}$ , $x^{1/18}$

Assumptions
1. We are given the expression $\left(x^{\frac{1}{3}}\right)\left(x^{\frac{1}{6}}\right)$ .
2. We need to simplify the expression to its most simplified form.

To simplify the expression, we will use the property of exponents that states when multiplying two exponents with the same base, we add the exponents.
$x^a \cdot x^b = x^{a+b}$

Apply the property of exponents to the given expression.
$\left(x^{\frac{1}{3}}\right)\left(x^{\frac{1}{6}}\right) = x^{\frac{1}{3} + \frac{1}{6}}$

Find a common denominator to add the fractions $\frac{1}{3}$ and $\frac{1}{6}$ .
The common denominator for 3 and 6 is 6.

Express $\frac{1}{3}$ with the common denominator 6.
$\frac{1}{3} = \frac{2}{6}$

Now, add the exponents with the common denominator.
$\frac{2}{6} + \frac{1}{6} = \frac{3}{6}$

Simplify the fraction $\frac{3}{6}$ .
$\frac{3}{6} = \frac{1}{2}$

Now, we can write the simplified exponent with the base $x$ .
$x^{\frac{3}{6}} = x^{\frac{1}{2}}$

The expression $x^{\frac{1}{2}}$ is equivalent to the square root of $x$ .
$x^{\frac{1}{2}} = \sqrt{x}$
The most simplified form of the expression $\left(x^{\frac{1}{3}}\right)\left(x^{\frac{1}{6}}\right)$ is $\sqrt{x}$ .