Simplify 2a^(1/2)(5a^(1/2)b^(1/4))^2

$2 a^{\frac{1}{2}} {(5 a^{\frac{1}{2}} b^{\frac{1}{4}})}^{2}$

Step 1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $5 a^{\frac{1}{2}} b^{\frac{1}{4}}$ .

$2 a^{\frac{1}{2}} ({(5 a^{\frac{1}{2}})}^{2} {(b^{\frac{1}{4}})}^{2})$

Apply the product rule to $5 a^{\frac{1}{2}}$ .

$2 a^{\frac{1}{2}} (5^{2} {(a^{\frac{1}{2}})}^{2} {(b^{\frac{1}{4}})}^{2})$

Step 2

Raise $5$ to the power of $2$ .

$2 a^{\frac{1}{2}} (25 {(a^{\frac{1}{2}})}^{2} {(b^{\frac{1}{4}})}^{2})$

Step 3

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

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

$2 a^{\frac{1}{2}} (25 a^{\frac{1}{2} \cdot 2} {(b^{\frac{1}{4}})}^{2})$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 a^{\frac{1}{2}} (25 a^{\frac{1}{2} \cdot 2} {(b^{\frac{1}{4}})}^{2})$

Rewrite the expression.

$2 a^{\frac{1}{2}} (25 a^{1} {(b^{\frac{1}{4}})}^{2})$

Step 4

Multiply $a^{\frac{1}{2}}$ by $a^{1}$ by adding the exponents.

Move $a^{1}$ .

$2 (a^{1} a^{\frac{1}{2}}) (25 {(b^{\frac{1}{4}})}^{2})$

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

$2 a^{1 + \frac{1}{2}} (25 {(b^{\frac{1}{4}})}^{2})$

Write $1$ as a fraction with a common denominator.

$2 a^{\frac{2}{2} + \frac{1}{2}} (25 {(b^{\frac{1}{4}})}^{2})$

Combine the numerators over the common denominator.

$2 a^{\frac{2 + 1}{2}} (25 {(b^{\frac{1}{4}})}^{2})$

Add $2$ and $1$ .

$2 a^{\frac{3}{2}} (25 {(b^{\frac{1}{4}})}^{2})$

Step 5

Rewrite using the commutative property of multiplication.

$2 \cdot 25 a^{\frac{3}{2}} {(b^{\frac{1}{4}})}^{2}$

Step 6

Multiply $2$ by $25$ .

$50 a^{\frac{3}{2}} {(b^{\frac{1}{4}})}^{2}$

Step 7

Multiply the exponents in ${(b^{\frac{1}{4}})}^{2}$ .

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

$50 a^{\frac{3}{2}} b^{\frac{1}{4} \cdot 2}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$50 a^{\frac{3}{2}} b^{\frac{1}{2 (2)} \cdot 2}$

Cancel the common factor.

$50 a^{\frac{3}{2}} b^{\frac{1}{2 \cdot 2} \cdot 2}$

Rewrite the expression.

$50 a^{\frac{3}{2}} b^{\frac{1}{2}}$

Simplify 2a^(1/2)(5a^(1/2)b^(1/4))^2

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