Algebra Examples

${(x^{\frac{1}{2}})}^{3} \sqrt{x^{3}} = x^{a}$

Step 1

Rewrite the equation as $x^{a} = {(x^{\frac{1}{2}})}^{3} \sqrt{x^{3}}$ .

$x^{a} = {(x^{\frac{1}{2}})}^{3} \sqrt{x^{3}}$

Step 2

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

$x^{a} = {(x^{\frac{1}{2}})}^{3} x^{\frac{3}{2}}$

Step 3

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

$x^{a} = x^{\frac{1}{2} \cdot 3} x^{\frac{3}{2}}$

Step 4

Multiply $x^{\frac{1}{2} \cdot 3}$ by $x^{\frac{3}{2}}$ by adding the exponents.

Tap for more steps...

Step 4.1

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

$x^{a} = x^{\frac{1}{2} \cdot 3 + \frac{3}{2}}$

Step 4.2

Combine $\frac{1}{2}$ and $3$ .

$x^{a} = x^{\frac{3}{2} + \frac{3}{2}}$

Step 4.3

Combine the numerators over the common denominator.

$x^{a} = x^{\frac{3 + 3}{2}}$

Step 4.4

Add $3$ and $3$ .

$x^{a} = x^{\frac{6}{2}}$

Step 4.5

Divide $6$ by $2$ .

$x^{a} = x^{3}$

Step 5

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

$a = 3$