Simplify the Radical Expression square root of a( square root of a^3-7)

$\sqrt{a} (\sqrt{a^{3}} - 7)$

Step 1

Simplify each term.

Factor out $a^{2}$ .

$\sqrt{a} (\sqrt{a^{2} a} - 7)$

Pull terms out from under the radical.

$\sqrt{a} (| a | \sqrt{a} - 7)$

Step 2

Apply the distributive property.

$\sqrt{a} (| a | \sqrt{a}) + \sqrt{a} \cdot - 7$

Step 3

Multiply $\sqrt{a} (| a | \sqrt{a})$ .

Raise $\sqrt{a}$ to the power of $1$ .

${\sqrt{a}}^{1} \sqrt{a} | a | + \sqrt{a} \cdot - 7$

Raise $\sqrt{a}$ to the power of $1$ .

${\sqrt{a}}^{1} {\sqrt{a}}^{1} | a | + \sqrt{a} \cdot - 7$

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

${\sqrt{a}}^{1 + 1} | a | + \sqrt{a} \cdot - 7$

Add $1$ and $1$ .

${\sqrt{a}}^{2} | a | + \sqrt{a} \cdot - 7$

Step 4

Move $- 7$ to the left of $\sqrt{a}$ .

${\sqrt{a}}^{2} | a | - 7 \cdot \sqrt{a}$

Step 5

Rewrite ${\sqrt{a}}^{2}$ as $a$ .

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

${(a^{\frac{1}{2}})}^{2} | a | - 7 \cdot \sqrt{a}$

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

$a^{\frac{1}{2} \cdot 2} | a | - 7 \cdot \sqrt{a}$

Combine $\frac{1}{2}$ and $2$ .

$a^{\frac{2}{2}} | a | - 7 \cdot \sqrt{a}$

Cancel the common factor of $2$ .

Cancel the common factor.

$a^{\frac{2}{2}} | a | - 7 \cdot \sqrt{a}$

Rewrite the expression.

$a^{1} | a | - 7 \cdot \sqrt{a}$

Simplify.

$a | a | - 7 \cdot \sqrt{a}$

$a | a | - 7 \sqrt{a}$