Exponent worksheet

1. Result: 2

1. Reduce the exponent to lowest terms by dividing top and bottom by 3

   $\frac{3}{3}$$=\frac{1}{1}$

2. A fraction exponent p/q means two moves: take the root named by the bottom number q, then raise it to the power on top, p (here the first root, then the power 1)

   $2^{\frac{1}{1}}$$={\left(2^{\frac{1}{1}}\right)}^1$

3. Take the root first (the first root of a number is the number itself)

   $2^{\frac{1}{1}}$$=2$

2. Result: 2.83

1. A fraction exponent p/q means two moves: take the root named by the bottom number q, then raise it to the power on top, p (here the 2nd root, then the power 3)

   $2^{\frac{3}{2}}$$={\left(2^{\frac{1}{2}}\right)}^3$

2. Take the root first: find the number that gives the base when 2 copies of it are multiplied together

   $2^{\frac{1}{2}}$$=1.4142135624$

3. Write down 3 copies of 1.4142135624 and multiply them together

   ${1.4142135624}^3$$=1.4142135624\times 1.4142135624\times 1.4142135624$$=2.8284271247$

3. Result: 1,024

1. Write down 10 copies of 2 and multiply them together

   $2^{10}$$=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2$$=\mathrm{1,024}$

4. Exponent (n): 9.64

1. Find the exponent n that makes 2^n equal 800

   $2^n$$=800$

2. Take the natural log (ln) of both sides. Logs turn a power into a multiplication, which brings n down where you can reach it

   $n\times \ln \left(2\right)$$=\ln \left(800\right)$

3. Divide both sides by the log of the base to get n on its own

   $n$$=\frac{\ln \left(800\right)}{\ln \left(2\right)}$$=\frac{6.6846117277}{0.6931471806}$$=9.6438561898$

5. Result: 1

1. Reduce the exponent to lowest terms by dividing top and bottom by 2

   $\frac{4}{2}$$=\frac{2}{1}$

2. A fraction exponent p/q means two moves: take the root named by the bottom number q, then raise it to the power on top, p (here the first root, then the power 2)

   $1^{\frac{2}{1}}$$={\left(1^{\frac{1}{1}}\right)}^2$

3. Take the root first (the first root of a number is the number itself)

   $1^{\frac{1}{1}}$$=1$

4. Write down 2 copies of 1 and multiply them together

   $1^2$$=1\times 1$$=1$

6. Exponent (n): 6.14

1. Find the exponent n that makes 3^n equal 850

   $3^n$$=850$

2. Take the natural log (ln) of both sides. Logs turn a power into a multiplication, which brings n down where you can reach it

   $n\times \ln \left(3\right)$$=\ln \left(850\right)$

3. Divide both sides by the log of the base to get n on its own

   $n$$=\frac{\ln \left(850\right)}{\ln \left(3\right)}$$=\frac{6.7452363495}{1.0986122887}$$=6.1397787182$

7. Exponent (n): 6.37

1. Find the exponent n that makes 3^n equal 1,100

   $3^n$$=\mathrm{1,100}$

2. Take the natural log (ln) of both sides. Logs turn a power into a multiplication, which brings n down where you can reach it

   $n\times \ln \left(3\right)$$=\ln \left(\mathrm{1,100}\right)$

3. Divide both sides by the log of the base to get n on its own

   $n$$=\frac{\ln \left(\mathrm{1,100}\right)}{\ln \left(3\right)}$$=\frac{7.0030654588}{1.0986122887}$$=6.3744648872$

8. Result: 2,048

1. Raise 2 to the power 11

   $2^{11}$$=\mathrm{2,048}$

9. Exponent (n): 9.97

1. Find the exponent n that makes 2^n equal 1,000

   $2^n$$=\mathrm{1,000}$

2. Take the natural log (ln) of both sides. Logs turn a power into a multiplication, which brings n down where you can reach it

   $n\times \ln \left(2\right)$$=\ln \left(\mathrm{1,000}\right)$

3. Divide both sides by the log of the base to get n on its own

   $n$$=\frac{\ln \left(\mathrm{1,000}\right)}{\ln \left(2\right)}$$=\frac{6.907755279}{0.6931471806}$$=9.9657842847$

10. Result: 1

1. A fraction exponent p/q means two moves: take the root named by the bottom number q, then raise it to the power on top, p (here the 2nd root, then the power 3)

   $1^{\frac{3}{2}}$$={\left(1^{\frac{1}{2}}\right)}^3$

2. Take the root first: find the number that gives the base when 2 copies of it are multiplied together

   $1^{\frac{1}{2}}$$=1$

3. Write down 3 copies of 1 and multiply them together

   $1^3$$=1\times 1\times 1$$=1$