Pre-Algebra Examples

$\sqrt[3]{b^{2} c^{2}} \sqrt{b c}$

Step 1

Multiply $\sqrt[3]{b^{2} c^{2}} \sqrt{b c}$ .

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{b^{2} c^{2}}$ as ${(b^{2} c^{2})}^{\frac{1}{3}}$ .

${(b^{2} c^{2})}^{\frac{1}{3}} \sqrt{b c}$

Step 1.1.2

Rewrite ${(b^{2} c^{2})}^{\frac{1}{3}}$ as ${(b^{2} c^{2})}^{\frac{2}{6}}$ .

${(b^{2} c^{2})}^{\frac{2}{6}} \sqrt{b c}$

Step 1.1.3

Rewrite ${(b^{2} c^{2})}^{\frac{2}{6}}$ as $\sqrt[6]{{(b^{2} c^{2})}^{2}}$ .

$\sqrt[6]{{(b^{2} c^{2})}^{2}} \sqrt{b c}$

Step 1.1.4

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

$\sqrt[6]{{(b^{2} c^{2})}^{2}} {(b c)}^{\frac{1}{2}}$

Step 1.1.5

Rewrite ${(b c)}^{\frac{1}{2}}$ as ${(b c)}^{\frac{3}{6}}$ .

$\sqrt[6]{{(b^{2} c^{2})}^{2}} {(b c)}^{\frac{3}{6}}$

Step 1.1.6

Rewrite ${(b c)}^{\frac{3}{6}}$ as $\sqrt[6]{{(b c)}^{3}}$ .

$\sqrt[6]{{(b^{2} c^{2})}^{2}} \sqrt[6]{{(b c)}^{3}}$

Step 1.2

Combine using the product rule for radicals.

$\sqrt[6]{{(b^{2} c^{2})}^{2} {(b c)}^{3}}$

Step 1.3

Apply the product rule to $b^{2} c^{2}$ .

$\sqrt[6]{{(b^{2})}^{2} {(c^{2})}^{2} {(b c)}^{3}}$

Step 1.4

Apply the product rule to $b c$ .

$\sqrt[6]{{(b^{2})}^{2} {(c^{2})}^{2} b^{3} c^{3}}$

Step 1.5

Multiply the exponents in ${(b^{2})}^{2}$ .

Tap for more steps...

Step 1.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt[6]{b^{2 \cdot 2} {(c^{2})}^{2} b^{3} c^{3}}$

Step 1.5.2

Multiply $2$ by $2$ .

$\sqrt[6]{b^{4} {(c^{2})}^{2} b^{3} c^{3}}$

Step 1.6

Multiply the exponents in ${(c^{2})}^{2}$ .

Tap for more steps...

Step 1.6.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt[6]{b^{4} c^{2 \cdot 2} b^{3} c^{3}}$

Step 1.6.2

Multiply $2$ by $2$ .

$\sqrt[6]{b^{4} c^{4} b^{3} c^{3}}$

Step 1.7

Combine exponents.

Tap for more steps...

Step 1.7.1

Multiply $b^{4}$ by $b^{3}$ by adding the exponents.

Tap for more steps...

Step 1.7.1.1

Move $b^{3}$ .

$\sqrt[6]{b^{3} b^{4} c^{4} c^{3}}$

Step 1.7.1.2

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

$\sqrt[6]{b^{3 + 4} c^{4} c^{3}}$

Step 1.7.1.3

Add $3$ and $4$ .

$\sqrt[6]{b^{7} c^{4} c^{3}}$

Step 1.7.2

Multiply $c^{4}$ by $c^{3}$ by adding the exponents.

Tap for more steps...

Step 1.7.2.1

Move $c^{3}$ .

$\sqrt[6]{b^{7} (c^{3} c^{4})}$

Step 1.7.2.2

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

$\sqrt[6]{b^{7} c^{3 + 4}}$

Step 1.7.2.3

Add $3$ and $4$ .

$\sqrt[6]{b^{7} c^{7}}$

Step 2

Rewrite $b^{7} c^{7}$ as ${(b c)}^{6} (b c)$ .

Tap for more steps...

Step 2.1

Factor out $b^{6}$ .

$\sqrt[6]{b^{6} b c^{7}}$

Step 2.2

Factor out $c^{6}$ .

$\sqrt[6]{b^{6} b (c^{6} c)}$

Step 2.3

Move $b$ .

$\sqrt[6]{b^{6} c^{6} b c}$

Step 2.4

Rewrite $b^{6} c^{6}$ as ${(b c)}^{6}$ .

$\sqrt[6]{{(b c)}^{6} b c}$

Step 2.5

Add parentheses.

$\sqrt[6]{{(b c)}^{6} (b c)}$

Step 3

Pull terms out from under the radical.

$b c \sqrt[6]{b c}$