Simplify square root of 25x^9* cube root of 27x^4

Solution:

Step:1 Express $25x^9$ as $(5x^4{)}^2\cdot x$.

Step:1.1 Represent $25$ as $5^2$. $\sqrt{5^2{x}^9}\cdot \sqrt[3]{27x^4}$

Step:1.2 Extract $x^8$. $\sqrt{5^2(x^8\cdot x)}\cdot \sqrt[3]{27x^4}$

Step:1.3 Express $x^8$ as $(x^4{)}^2$. $\sqrt{5^2((x^4{)}^2\cdot x)}\cdot \sqrt[3]{27x^4}$

Step:1.4 Represent $5^2(x^4{)}^2$ as $(5x^4{)}^2$. $\sqrt{(5x^4{)}^2\cdot x}\cdot \sqrt[3]{27x^4}$

Step:2 Extract terms from under the radical. $5x^4\sqrt{x}\cdot \sqrt[3]{27x^4}$

Step:3 Express $27x^4$ as $(3x{)}^3\cdot x$.

Step:3.1 Represent $27$ as $3^3$. $5x^4\sqrt{x}\cdot \sqrt[3]{3^3{x}^4}$

Step:3.2 Extract $x^3$. $5x^4\sqrt{x}\cdot \sqrt[3]{3^3(x^3\cdot x)}$

Step:3.3 Express $3^3{x}^3$ as $(3x{)}^3$. $5x^4\sqrt{x}\cdot \sqrt[3]{(3x{)}^3\cdot x}$

Step:4 Extract terms from under the radical. $5x^4\sqrt{x}\cdot (3x\sqrt[3]{x})$

Step:5 Combine $x^4$ and $x$ by summing their exponents.

Step:5.1 Rearrange $x$. $5(x\cdot x^4)\sqrt{x}\cdot (3\sqrt[3]{x})$

Step:5.2 Multiply $x$ by $x^4$.

Step:5.2.1 Raise $x$ to the power of $1$. $5(x^1{x}^4)\sqrt{x}\cdot (3\sqrt[3]{x})$

Step:5.2.2 Apply the power rule $a^m{a}^n=a^{m+n}$ to combine exponents. $5x^{1+4}\sqrt{x}\cdot (3\sqrt[3]{x})$

Step:5.3 Add $1$ and $4$. $5x^5\sqrt{x}\cdot (3\sqrt[3]{x})$

Step:6 Multiply $5x^5\sqrt{x}(3\sqrt[3]{x})$.

Step:6.1 Multiply $3$ by $5$. $15x^5\sqrt{x}\sqrt[3]{x}$

Step:6.2 Express the expression using the least common index of $6$.

Step:6.2.1 Use $\sqrt[n]{a^x}=a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$. $15x^5(x^{\frac{1}{3}}\sqrt{x})$

Step:6.2.2 Express $x^{\frac{1}{3}}$ as $x^{\frac{2}{6}}$. $15x^5(x^{\frac{2}{6}}\sqrt{x})$

Step:6.2.3 Represent $x^{\frac{2}{6}}$ as $\sqrt[6]{x^2}$. $15x^5(\sqrt[6]{x^2}\sqrt{x})$

Step:6.2.4 Use $\sqrt[n]{a^x}=a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$. $15x^5(\sqrt[6]{x^2}{x}^{\frac{1}{2}})$

Step:6.2.5 Express $x^{\frac{1}{2}}$ as $x^{\frac{3}{6}}$. $15x^5(\sqrt[6]{x^2}{x}^{\frac{3}{6}})$

Step:6.2.6 Represent $x^{\frac{3}{6}}$ as $\sqrt[6]{x^3}$. $15x^5(\sqrt[6]{x^2}\sqrt[6]{x^3})$

Step:6.3 Combine using the product rule for radicals. $15x^5\sqrt[6]{x^2{x}^3}$

Step:6.4 Multiply $x^2$ by $x^3$ by adding the exponents.

Step:6.4.1 Apply the power rule $a^m{a}^n=a^{m+n}$ to combine exponents. $15x^5\sqrt[6]{x^{2+3}}$

Step:6.4.2 Add $2$ and $3$. $15x^5\sqrt[6]{x^5}$

Knowledge Notes:

The problem involves simplifying an expression that contains both square and cube roots. To simplify such an expression, we can use several algebraic rules and properties of exponents and radicals:

1. Square Root: $\sqrt{x^2}=x$. The square root of a squared number is the number itself.

2. Cube Root: $\sqrt[3]{x^3}=x$. The cube root of a cubed number is the number itself.

3. Power Rule: $a^m\cdot a^n=a^{m+n}$. When multiplying like bases, you add the exponents.

4. Radical Multiplication: $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{ab}$. You can multiply under one radical if they have the same index.

5. Rewriting Radicals: $\sqrt[n]{a^x}=a^{\frac{x}{n}}$. A radical can be rewritten as an exponent by dividing the exponent by the index of the radical.

6. Least Common Index: When dealing with multiple radicals, it's often helpful to rewrite them with the same index to combine them more easily.

In the given solution, these rules are applied step by step to simplify the expression. The square root and cube root are first simplified by extracting perfect squares and cubes, respectively. Then, the expression is further simplified by combining like terms and using the properties of exponents to combine and simplify the expression to its simplest form.