Algebra Examples

${(3 x^{3})}^{2} \cdot {(25 x^{6})}^{\frac{1}{2}}$

Step 1

Simplify the expression.

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Step 1.1

Apply the product rule to $3 x^{3}$ .

$3^{2} {(x^{3})}^{2} \cdot {(25 x^{6})}^{\frac{1}{2}}$

Step 1.2

Raise $3$ to the power of $2$ .

$9 {(x^{3})}^{2} \cdot {(25 x^{6})}^{\frac{1}{2}}$

Step 1.3

Multiply the exponents in ${(x^{3})}^{2}$ .

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Step 1.3.1

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

$9 x^{3 \cdot 2} \cdot {(25 x^{6})}^{\frac{1}{2}}$

Step 1.3.2

Multiply $3$ by $2$ .

$9 x^{6} \cdot {(25 x^{6})}^{\frac{1}{2}}$

Step 1.4

Apply the product rule to $25 x^{6}$ .

$9 x^{6} \cdot (25^{\frac{1}{2}} {(x^{6})}^{\frac{1}{2}})$

Step 1.5

Rewrite using the commutative property of multiplication.

$9 \cdot 25^{\frac{1}{2}} x^{6} {(x^{6})}^{\frac{1}{2}}$

Step 1.6

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

$9 \cdot {(5^{2})}^{\frac{1}{2}} x^{6} {(x^{6})}^{\frac{1}{2}}$

Step 1.7

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

$9 \cdot 5^{2 (\frac{1}{2})} x^{6} {(x^{6})}^{\frac{1}{2}}$

Step 2

Cancel the common factor of $2$ .

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Step 2.1

Cancel the common factor.

$9 \cdot 5^{2 (\frac{1}{2})} x^{6} {(x^{6})}^{\frac{1}{2}}$

Step 2.2

Rewrite the expression.

$9 \cdot 5^{1} x^{6} {(x^{6})}^{\frac{1}{2}}$

Step 3

Evaluate the exponent.

$9 \cdot 5 x^{6} {(x^{6})}^{\frac{1}{2}}$

Step 4

Multiply $9$ by $5$ .

$45 x^{6} {(x^{6})}^{\frac{1}{2}}$

Step 5

Multiply the exponents in ${(x^{6})}^{\frac{1}{2}}$ .

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Step 5.1

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

$45 x^{6} x^{6 (\frac{1}{2})}$

Step 5.2

Cancel the common factor of $2$ .

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Step 5.2.1

Factor $2$ out of $6$ .

$45 x^{6} x^{2 (3) \frac{1}{2}}$

Step 5.2.2

Cancel the common factor.

$45 x^{6} x^{2 \cdot 3 \frac{1}{2}}$

Step 5.2.3

Rewrite the expression.

$45 x^{6} x^{3}$

Step 6

Multiply $x^{6}$ by $x^{3}$ by adding the exponents.

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Step 6.1

Move $x^{3}$ .

$45 (x^{3} x^{6})$

Step 6.2

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

$45 x^{3 + 6}$

Step 6.3

Add $3$ and $6$ .

$45 x^{9}$