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

$\sqrt{25 x^{9}} \cdot \sqrt[3]{27 x^{4}}$

Step 1

Rewrite $25 x^{9}$ as ${(5 x^{4})}^{2} x$ .

Rewrite $25$ as $5^{2}$ .

$\sqrt{5^{2} x^{9}} \cdot \sqrt[3]{27 x^{4}}$

Factor out $x^{8}$ .

$\sqrt{5^{2} (x^{8} x)} \cdot \sqrt[3]{27 x^{4}}$

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

$\sqrt{5^{2} ({(x^{4})}^{2} x)} \cdot \sqrt[3]{27 x^{4}}$

Rewrite $5^{2} {(x^{4})}^{2}$ as ${(5 x^{4})}^{2}$ .

$\sqrt{{(5 x^{4})}^{2} x} \cdot \sqrt[3]{27 x^{4}}$

Step 2

Pull terms out from under the radical.

$5 x^{4} \sqrt{x} \cdot \sqrt[3]{27 x^{4}}$

Step 3

Rewrite $27 x^{4}$ as ${(3 x)}^{3} x$ .

Rewrite $27$ as $3^{3}$ .

$5 x^{4} \sqrt{x} \cdot \sqrt[3]{3^{3} x^{4}}$

Factor out $x^{3}$ .

$5 x^{4} \sqrt{x} \cdot \sqrt[3]{3^{3} (x^{3} x)}$

Rewrite $3^{3} x^{3}$ as ${(3 x)}^{3}$ .

$5 x^{4} \sqrt{x} \cdot \sqrt[3]{{(3 x)}^{3} x}$

Step 4

Pull terms out from under the radical.

$5 x^{4} \sqrt{x} \cdot (3 x \sqrt[3]{x})$

Step 5

Multiply $x^{4}$ by $x$ by adding the exponents.

Move $x$ .

$5 (x \cdot x^{4}) \sqrt{x} \cdot (3 \sqrt[3]{x})$

Multiply $x$ by $x^{4}$ .

Raise $x$ to the power of $1$ .

$5 (x^{1} x^{4}) \sqrt{x} \cdot (3 \sqrt[3]{x})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5 x^{1 + 4} \sqrt{x} \cdot (3 \sqrt[3]{x})$

Add $1$ and $4$ .

$5 x^{5} \sqrt{x} \cdot (3 \sqrt[3]{x})$

Step 6

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

Multiply $3$ by $5$ .

$15 x^{5} \sqrt{x} \sqrt[3]{x}$

Rewrite the expression using the least common index of $6$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$15 x^{5} (x^{\frac{1}{3}} \sqrt{x})$

Rewrite $x^{\frac{1}{3}}$ as $x^{\frac{2}{6}}$ .

$15 x^{5} (x^{\frac{2}{6}} \sqrt{x})$

Rewrite $x^{\frac{2}{6}}$ as $\sqrt[6]{x^{2}}$ .

$15 x^{5} (\sqrt[6]{x^{2}} \sqrt{x})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$15 x^{5} (\sqrt[6]{x^{2}} x^{\frac{1}{2}})$

Rewrite $x^{\frac{1}{2}}$ as $x^{\frac{3}{6}}$ .

$15 x^{5} (\sqrt[6]{x^{2}} x^{\frac{3}{6}})$

Rewrite $x^{\frac{3}{6}}$ as $\sqrt[6]{x^{3}}$ .

$15 x^{5} (\sqrt[6]{x^{2}} \sqrt[6]{x^{3}})$

Combine using the product rule for radicals.

$15 x^{5} \sqrt[6]{x^{2} x^{3}}$

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$15 x^{5} \sqrt[6]{x^{2 + 3}}$

Add $2$ and $3$ .

$15 x^{5} \sqrt[6]{x^{5}}$

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

Related Questions

Support LCMGCF.com by disabling your adblocker.