Question

Calculate these expressions in your head. Use the Law of Exponents to help you.

1. $\frac{{\mathbf{18}}^{\mathbf{5}}}{{\mathbf{9}}^{\mathbf{5}}}$
2. ${\mathbf{20}}^{\mathbf{6}}\mathbf{\cdot }{\mathbf{(}\mathbf{0}\mathbf{.5}\mathbf{)}}^{\mathbf{6}}$
   

Short answer

1. $32$
   
2. ${10}^6$
   

Step-by-step solution

Part a. Step 1. Exponent rule.

Power of fraction rule:${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}$

Part a. Step 2. Rewrite the expression.

Use the power of the fraction rule.

$\begin{array}{l}\frac{{18}^5}{9^5}={\left(\frac{18}{9}\right)}^5\\ =2^5\end{array}$

Part a. Step 3. Solve the exponent.

$\begin{array}{l}\frac{{18}^5}{9^5}=2^5\\ =32\end{array}$

Part b. Step 1. Exponent rule.

Power rule:${\left(ab\right)}^n=a^n{b}^n$

Part b. Step 2. Rewrite the expression.

Use the power rule.

${20}^6\cdot {\left(0.5\right)}^6={\left(20\cdot \left(0.5\right)\right)}^6$

Part b. Step 3. Simplify.

${20}^6\cdot {\left(0.5\right)}^6={10}^6$