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

Solution:

Step:1

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

Step:1.1

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

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

Step:1.2

Rearrange to place $x$ outside.

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

Step:1.3

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

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

Step:1.4

Enclose with parentheses.

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

Step:2

Extract terms from under the square root.

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

Step:3

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

Step:3.1

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

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

Step:3.2

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

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

Step:3.3

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

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

Step:3.4

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

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

Step:3.5

Surround with parentheses.

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

Step:4

Extract terms from under the radical.

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

Step:5

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

Step:5.1

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

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

Step:5.2

Calculate $4\times 2$.

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

Step:6

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

Step:6.1

Merge using the product property of square roots.

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

Step:6.2

Multiply $30$ by $10$.

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

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 $a^m{a}^n=a^{m+n}$.

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

Step:6.6

Add $1$ and $1$.

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

Step:7

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

Step:7.1

Extract $100$ from $300$.

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

Step:7.2

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

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

Step:7.3

Reposition $3$.

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

Step:7.4

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

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

Step:7.5

Enclose with parentheses.

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

Step:8

Extract terms from under the radical.

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

Step:9

Reorder using the commutative property.

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

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.

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

Step:10.1.2

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

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

Step:10.2

Sum $8$ and $1$.

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

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.