Question

In the following exercises, simplify each expression by applying several properties.

$$\begin{gathered} \left(a\right).{\left(5t^2\right)}^3{\left(3t\right)}^2 \\ \left(b\right).\frac{{\left(t^2\right)}^5{\left(t^{-4}\right)}^2}{{\left(t^3\right)}^7} \\ \left(c\right).{\left(\frac{2x{y}^2}{x^3{y}^{-2}}\right)}^2{\left(\frac{12x{y}^3}{x^3{y}^{-1}}\right)}^{-1} \end{gathered}$$

Short answer

The simplified form of the expressions

$$\begin{gathered} \left(a\right).{\left(5t^2\right)}^3{\left(3t\right)}^2=1125t^8 \\ \left(b\right).\frac{{\left(t^2\right)}^5{\left(t^{-4}\right)}^2}{{\left(t^3\right)}^7}=\frac{1}{t^{19}} \\ \left(c\right){\left(\frac{2x{y}^2}{x^3{y}^{-2}}\right)}^2{\left(\frac{12x{y}^3}{x^3{y}^{-1}}\right)}^{-1}=\frac{y^4}{3x^2} \end{gathered}$$

Step-by-step solution

Part (a) Step 1. Given information.

The given information is${\left(5t^2\right)}^3{\left(3t\right)}^2$.

Part (a) Step 2. Calculate the expression.

Calculation of expression:

First of all, apply the product to a power property.

We get,

Now, apply power property.

We get,

Part (b) Given information.

The given expression is$\frac{{\left(t^2\right)}^5{\left(t^{-4}\right)}^2}{{\left(t^3\right)}^7}$.

Part (b) Step 2. Calculate the expression.

First of all, apply power property-

We get,

Now, apply property of negative exponents and cancel out the common terms.

We get,

Part (c) Given information.

The given expression is${\left(\frac{2x{y}^2}{x^3{y}^{-2}}\right)}^2{\left(\frac{12x{y}^3}{x^3{y}^{-1}}\right)}^{-1}$.

Part (c) Step 2. Calculate the expression.

First of all, apply the product to a power property.

We get,

Now, apply power property.

We get,

Finally, apply property of negative exponents and cancel out the common terms.

We get,