Question1. Simplify with positive exponents: a) $\sqrt{\frac{(25 x)^{\frac{1}{2}}\left(8 x^{2}\right)^{\frac{1}{3}}}{40 x^{\frac{1}{6}}}}$ b) $\frac{\left(8 x^{3} y^{4}\right)^{2}\left(-25 x^{3} y^{2}\right)^{3}}{\left(-10 x^{3} y^{2}\right)^{6}}$

Studdy Solution

STEP 1

Assumptions1. All variables are real numbers. . The square root of a number is always positive.
3. The division by zero is undefined.

STEP 2

First, let's simplify the expression a) $\sqrt{\frac{(25 x)^{\frac{1}{2}}\left(8 x^{2}\right)^{\frac{1}{}}}{40 x^{\frac{1}{6}}}}$ by distributing the exponents inside the brackets.
$\sqrt{\frac{(25^{\frac{1}{2}} x^{\frac{1}{2}})(8^{\frac{1}{}} x^{\frac{2}{}})}{40^{\frac{1}{6}} x^{\frac{1}{6}}}}$

STEP 3

Now, simplify the exponents inside the square root.
$\sqrt{\frac{(5 x^{\frac{1}{2}})(2 x^{\frac{2}{3}})}{2^{\frac{1}{3}} x^{\frac{1}{6}}}}$

STEP 4

Multiply the terms with the same base inside the square root.
$\sqrt{\frac{10 x^{\frac{7}{6}}}{2^{\frac{1}{3}} x^{\frac{1}{6}}}}$

STEP 5

implify the terms with the same base inside the square root.
$\sqrt{5 x}$

STEP 6

Let's simplify the expression b) $\frac{\left(8 x^{3} y^{4}\right)^{2}\left(-25 x^{3} y^{2}\right)^{3}}{\left(-10 x^{3} y^{2}\right)^{6}}$ by distributing the exponents inside the brackets.
$\frac{(8^{2} x^{6} y^{8})(-25^{3} x^{9} y^{6})}{(-10^{6} x^{18} y^{12})}$

STEP 7

Now, simplify the exponents.
$\frac{(64 x^{6} y^{})(-15625 x^{9} y^{6})}{(-1000000 x^{18} y^{12})}$

STEP 8

Multiply the terms with the same base.
$\frac{-1000000 x^{15} y^{14}}{-1000000 x^{18} y^{12}}$

STEP 9

implify the terms with the same base.
$x^{-3} y^{2}$

STEP 10

Express the final answer with positive exponents only.
$\frac{y^{2}}{x^{3}}$ So, the simplified expressions are $\sqrt{5 x}$ and $\frac{y^{2}}{x^{3}}$ .

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Question1. Simplify with positive exponents: a) $\sqrt{\frac{(25 x)^{\frac{1}{2}}\left(8 x^{2}\right)^{\frac{1}{3}}}{40 x^{\frac{1}{6}}}}$ b) $\frac{\left(8 x^{3} y^{4}\right)^{2}\left(-25 x^{3} y^{2}\right)^{3}}{\left(-10 x^{3} y^{2}\right)^{6}}$

Studdy Solution

STEP 1

Assumptions1. All variables are real numbers. . The square root of a number is always positive.
3. The division by zero is undefined.

STEP 2

First, let's simplify the expression a) $\sqrt{\frac{(25 x)^{\frac{1}{2}}\left(8 x^{2}\right)^{\frac{1}{}}}{40 x^{\frac{1}{6}}}}$ by distributing the exponents inside the brackets.
$\sqrt{\frac{(25^{\frac{1}{2}} x^{\frac{1}{2}})(8^{\frac{1}{}} x^{\frac{2}{}})}{40^{\frac{1}{6}} x^{\frac{1}{6}}}}$

STEP 3

Now, simplify the exponents inside the square root.
$\sqrt{\frac{(5 x^{\frac{1}{2}})(2 x^{\frac{2}{3}})}{2^{\frac{1}{3}} x^{\frac{1}{6}}}}$

STEP 4

Multiply the terms with the same base inside the square root.
$\sqrt{\frac{10 x^{\frac{7}{6}}}{2^{\frac{1}{3}} x^{\frac{1}{6}}}}$

STEP 5

implify the terms with the same base inside the square root.
$\sqrt{5 x}$

STEP 6

Let's simplify the expression b) $\frac{\left(8 x^{3} y^{4}\right)^{2}\left(-25 x^{3} y^{2}\right)^{3}}{\left(-10 x^{3} y^{2}\right)^{6}}$ by distributing the exponents inside the brackets.
$\frac{(8^{2} x^{6} y^{8})(-25^{3} x^{9} y^{6})}{(-10^{6} x^{18} y^{12})}$

STEP 7

Now, simplify the exponents.
$\frac{(64 x^{6} y^{})(-15625 x^{9} y^{6})}{(-1000000 x^{18} y^{12})}$

STEP 8

Multiply the terms with the same base.
$\frac{-1000000 x^{15} y^{14}}{-1000000 x^{18} y^{12}}$

STEP 9

implify the terms with the same base.
$x^{-3} y^{2}$

STEP 10

Express the final answer with positive exponents only.
$\frac{y^{2}}{x^{3}}$ So, the simplified expressions are $\sqrt{5 x}$ and $\frac{y^{2}}{x^{3}}$ .

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Studdy Solution

STEP 1

Assumptions1. All variables are real numbers. . The square root of a number is always positive.3. The division by zero is undefined.

STEP 2

STEP 3

Now, simplify the exponents inside the square root.(5x12)(2x23)213x16\sqrt{\frac{(5 x^{\frac{1}{2}})(2 x^{\frac{2}{3}})}{2^{\frac{1}{3}} x^{\frac{1}{6}}}}231​x61​(5x21​)(2x32​)​​

STEP 4

Multiply the terms with the same base inside the square root.10x76213x16\sqrt{\frac{10 x^{\frac{7}{6}}}{2^{\frac{1}{3}} x^{\frac{1}{6}}}}231​x61​10x67​​​

STEP 5

implify the terms with the same base inside the square root.5x\sqrt{5 x}5x​

STEP 6

STEP 7

Now, simplify the exponents.(64x6y)(−15625x9y6)(−1000000x18y12)\frac{(64 x^{6} y^{})(-15625 x^{9} y^{6})}{(-1000000 x^{18} y^{12})}(−1000000x18y12)(64x6y)(−15625x9y6)​

STEP 8

Multiply the terms with the same base.−1000000x15y14−1000000x18y12\frac{-1000000 x^{15} y^{14}}{-1000000 x^{18} y^{12}}−1000000x18y12−1000000x15y14​

STEP 9

implify the terms with the same base.x−3y2x^{-3} y^{2}x−3y2

STEP 10

Express the final answer with positive exponents only.y2x3\frac{y^{2}}{x^{3}}x3y2​So, the simplified expressions are 5x\sqrt{5 x}5x​ and y2x3\frac{y^{2}}{x^{3}}x3y2​.

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Assumptions1. All variables are real numbers. . The square root of a number is always positive.
3. The division by zero is undefined.

Now, simplify the exponents inside the square root.
$\sqrt{\frac{(5 x^{\frac{1}{2}})(2 x^{\frac{2}{3}})}{2^{\frac{1}{3}} x^{\frac{1}{6}}}}$

Multiply the terms with the same base inside the square root.
$\sqrt{\frac{10 x^{\frac{7}{6}}}{2^{\frac{1}{3}} x^{\frac{1}{6}}}}$

implify the terms with the same base inside the square root.
$\sqrt{5 x}$

Now, simplify the exponents.
$\frac{(64 x^{6} y^{})(-15625 x^{9} y^{6})}{(-1000000 x^{18} y^{12})}$

Multiply the terms with the same base.
$\frac{-1000000 x^{15} y^{14}}{-1000000 x^{18} y^{12}}$

implify the terms with the same base.
$x^{-3} y^{2}$

Express the final answer with positive exponents only.
$\frac{y^{2}}{x^{3}}$ So, the simplified expressions are $\sqrt{5 x}$ and $\frac{y^{2}}{x^{3}}$ .