Simplify square root of 10xz^10( square root of 30x^17z)

Solution:

Step:1

Express $10 x z^{10}$ as the product of $z^{5}$ squared and $10 x$.

Step:1.1

Represent $z^{10}$ as $(z^{5})^{2}$.

$\sqrt{10 x (z^{5})^{2}} \sqrt{30 x^{17} z}$

Step:1.2

Rearrange to place $x$ outside.

$\sqrt{10 (z^{5})^{2} x} \sqrt{30 x^{17} z}$

Step:1.3

Switch the positions of $10$ and $(z^{5})^{2}$.

$\sqrt{(z^{5})^{2} \cdot 10 x} \sqrt{30 x^{17} z}$

Step:1.4

Enclose with parentheses.

$\sqrt{(z^{5})^{2} \cdot (10 x)} \sqrt{30 x^{17} z}$

Step:2

Extract terms from under the square root.

$z^{5} \sqrt{10 x} \sqrt{30 x^{17} z}$

Step:3

Decompose $30 x^{17} z$ into a perfect square and remaining factors.

Step:3.1

Factor out $x^{16}$ from the expression.

$z^{5} \sqrt{10 x} \sqrt{30 x^{16} x z}$

Step:3.2

Express $x^{16}$ as $(x^{8})^{2}$.

$z^{5} \sqrt{10 x} \sqrt{30 (x^{8})^{2} x z}$

Step:3.3

Reorder $30$ and $(x^{8})^{2}$.

$z^{5} \sqrt{10 x} \sqrt{(x^{8})^{2} \cdot 30 x z}$

Step:3.4

Rewrite $x^{8}$ as $(x^{4})^{2}$.

$z^{5} \sqrt{10 x} \sqrt{((x^{4})^{2})^{2} \cdot 30 x z}$

Step:3.5

Surround with parentheses.

$z^{5} \sqrt{10 x} \sqrt{((x^{4})^{2})^{2} \cdot (30 x z)}$

Step:4

Extract terms from under the radical.

$z^{5} \sqrt{10 x} (x^{8} \sqrt{30 x z})$

Step:5

Apply exponent multiplication in $(x^{4})^{2}$.

Step:5.1

Utilize the power rule $(a^{m})^{n} = a^{mn}$.

$z^{5} \sqrt{10 x} (x^{4 \cdot 2} \sqrt{30 x z})$

Step:5.2

Calculate $4 \times 2$.

$z^{5} \sqrt{10 x} (x^{8} \sqrt{30 x z})$

Step:6

Combine $z^{5} \sqrt{10 x} (x^{8} \sqrt{30 x z})$.

Step:6.1

Merge using the product property of square roots.

$z^{5} (x^{8} \sqrt{10 x (30 x z)})$

Step:6.2

Multiply $30$ by $10$.

$z^{5} (x^{8} \sqrt{300 x (x z)})$

Step:6.3

Elevate $x$ to the first power.

$z^{5} (x^{8} \sqrt{300 (x^{1} x) z})$

Step:6.4

Raise $x$ to the first power again.

$z^{5} (x^{8} \sqrt{300 (x^{1} x^{1}) z})$

Step:6.5

Combine exponents using $a^{m} a^{n} = a^{m + n}$.

$z^{5} (x^{8} \sqrt{300 x^{1 + 1} z})$

Step:6.6

Add $1$ and $1$.

$z^{5} (x^{8} \sqrt{300 x^{2} z})$

Step:7

Express $300 x^{2} z$ as the square of $10 x$ times $3 z$.

Step:7.1

Extract $100$ from $300$.

$z^{5} (x^{8} \sqrt{100 (3) x^{2} z})$

Step:7.2

Represent $100$ as $10^{2}$.

$z^{5} (x^{8} \sqrt{(10)^{2} \cdot 3 x^{2} z})$

Step:7.3

Reposition $3$.

$z^{5} (x^{8} \sqrt{(10)^{2} x^{2} \cdot 3 z})$

Step:7.4

Rewrite $(10)^{2} x^{2}$ as $(10 x)^{2}$.

$z^{5} (x^{8} \sqrt{((10 x)^{2}) \cdot 3 z})$

Step:7.5

Enclose with parentheses.

$z^{5} (x^{8} \sqrt{((10 x)^{2}) \cdot (3 z)})$

Step:8

Extract terms from under the radical.

$z^{5} (x^{8} (10 x \sqrt{3 z}))$

Step:9

Reorder using the commutative property.

$10 z^{5} (x^{8} x \sqrt{3 z})$

Step:10

Add exponents when multiplying $x^{8}$ and $x$.

Step:10.1

Multiply $x^{8}$ by $x$.

Step:10.1.1

Raise $x$ to the first power.

$10 z^{5} (x^{8} x^{1} \sqrt{3 z})$

Step:10.1.2

Combine exponents with $a^{m} a^{n} = a^{m + n}$.

$10 z^{5} (x^{8 + 1} \sqrt{3 z})$

Step:10.2

Sum $8$ and $1$.

$10 z^{5} (x^{9} \sqrt{3 z})$ $10 z^{5} x^{9} \sqrt{3 z}$

Knowledge Notes:

1. Square Roots: The square root of a number is a value that, when multiplied by itself, gives the original number. The square root of $a$ is written as $\sqrt{a}$.

2. Exponent Rules: When an exponent is raised to another exponent, the powers are multiplied: $(a^{m})^{n} = a^{mn}$. When multiplying like bases, the exponents are added: $a^{m} a^{n} = a^{m + n}$.

3. Radicals: Terms under a square root can often be simplified by factoring out perfect squares.

4. Commutative Property: The order of multiplication does not affect the product: $ab = ba$.

5. Simplifying Expressions: Combining like terms and using algebraic rules to simplify expressions is a common practice in algebra.

6. LaTeX Formatting: Mathematical expressions can be neatly formatted using LaTeX, a typesetting system that uses commands to represent various mathematical symbols and structures.