Question

Simplify each expression.

${\left[{\left(\mathbf{2}{r}^{3}t\right)}^3\right]}^2$

Short answer

The expression ${\left[{\left(2r^{3}t\right)}^3\right]}^2$can be simplified as $\mathbf{64}{r}^{18}{t}^6$.

Step-by-step solution

Step 1. Write property of power of a product.

For any real number x, y and any integer n, ${\left(xy\right)}^n=x^n{y}^n$.

Step 2. Apply the property of power of a product.

Apply the property of power of a product to the given expression.

${\left(2^3\right)}^2{\left[{\left(r^3\right)}^3\right]}^2{\left(t^3\right)}^2$

Step 3. Write the property of power of a power.

For any real number a and any integers m and n,${\left(a^m\right)}^n=a^{m\cdot n}$

Step 4. Apply the property of power of a power.

Apply the property of power of a power to the expression in step 2.

$2^{3\cdot 2}{r}^{9\cdot 2}{t}^{3\cdot 2}$

Step 5. Simplify the obtained expression.

Simplify the expression in step.

$64r^{18}{t}^6$

Therefore,${\left[{\left(2r^{3}t\right)}^3\right]}^2$ can be simplified to $64r^{18}{t}^6$.