Properties of Rational Exponents and Radicals

Properties of Radicals

LT: I can simplify algebraic expression with radicals and rational exponents.

How do I simplify a radical?

Multiple Choice

Simplify $$\sqrt{700}$$ , if

$$\sqrt{700}=\sqrt{2\cdot2\cdot5\cdot5\cdot7}$$  

• $$2\sqrt{35}$$  

• $$7\sqrt{7}$$  

• $$100\sqrt{7}$$  

• $$10\sqrt{7}$$  

• $$70\sqrt{10}$$  

Multiple Choice

What is the prime factorization of 360?

• $$36\cdot10$$

• $$2\cdot180$$
  

• $$2\cdot2\cdot5\cdot18$$

• $$2\cdot2\cdot2\cdot3\cdot3\cdot5$$

• 360 is prime.

Multiple Choice

Simplify:

$$\sqrt{360}$$  

• $$5\sqrt{5}$$  
  

• $$5\sqrt{10}$$  
  

• $$72\sqrt{5}$$  

• $$6\sqrt{5}$$  

• $$6\sqrt{10}$$  

Multiple Choice

Simplify:

$$^3\sqrt{360}$$  

• $$6\ \cdot\ ^3\sqrt{5}$$  
  

• $$2\ \cdot\ ^3\sqrt{45}$$  
  

• $$3\ \cdot\ ^3\sqrt{40}$$  

• $$5\ \cdot\ ^3\sqrt{5}$$  

• None of the above

Multiple Choice

Simplify:
$$\sqrt{8x^3}$$  

• $$2\sqrt{x^3}$$  

• $$2\sqrt{2x^3}$$  

• $$2x\sqrt{2x}$$  

• $$x\sqrt{8x}$$  

• $$2x\sqrt{2}$$  

What is a rational exponent?

Multiple Choice

Rewrite using radical form:

$$5^{\frac{1}{2}}$$  

• $$\sqrt{5}$$  

• $$\frac{5}{2}$$  

• $$\left(\frac{1}{2}\right)^5$$  

• $$5^2$$  
  

• $$5\sqrt{2}$$  

Multiple Choice

Rewrite using radical form:

$$13^{\frac{1}{3}}$$  

• $$^3\sqrt{13}$$  

• $$\frac{13}{3}$$  

• $$\left(\frac{1}{3}\right)^{13}$$  

• $$13^3$$  

• $$13\sqrt{3}$$  

Multiple Choice

Simplify:

$$36^{\frac{1}{2}}$$  

• $$6$$  

• $$1296$$  

• $$18$$  

• $$72$$  

Multiple Choice

Simplify:

$$81^{\frac{1}{4}}$$   

• $$9$$  

• $$3$$  

• $$21.25$$  
  

• $$72$$  

Multiple Select

Select all answers that are equivalent to:

$$4^{\frac{1}{2}}$$  

• $$\sqrt{4}$$  
  

• $$2$$  
  

• $$4^2$$  

• $$\sqrt{2^2}$$  

How do I convert between radicals and rational exponents?

Conversion formula

•  $x^{\frac{m}{n}}=\ \ ^n\sqrt{x^m}$  

•  $x^{\frac{m}{n}}=\left(\ ^n\sqrt{x}\right)^m$  

Multiple Choice

Which expression is equivalent to:

$$\ ^5\sqrt{2^4}$$  

• $$5^{\frac{2}{4}}$$  

• $$2^{\frac{4}{5}}$$  

• $$2^{\frac{5}{4}}$$  

• $$4^{\frac{5}{2}}$$  

• $$4^{\frac{2}{5}}$$  

Multiple Choice

Which expression is equivalent to:

$$\ ^9\sqrt{x^7}$$  

• $$9x^7$$  
  

• $$x^{\frac{9}{7}}$$  
  

• $$9^{\frac{7}{x}}$$  

• $$9^{\frac{x}{7}}$$  
  

• $$x^{\frac{7}{9}}$$  
  

Multiple Select

Select all expressions that are equivalent to:

$$5^{\frac{3}{4}}$$  

• $$\left(\ ^4\sqrt{5}\right)^3$$  
  

• $$\left(\ ^4\sqrt{3}\right)^5$$  
  

• $$\ ^4\sqrt{3^5}$$  
  

• $$\ ^4\sqrt{5^3}$$  
  

Multiple Select

Select all expressions that are equivalent to:

$$x^{\frac{5}{2}}$$  

• $$\sqrt{x^5}$$  

• $$\ ^5\sqrt{x^2}$$  
  

• $$\left(\sqrt{x}\right)^5$$  

• $$\left(^{\ 5}\sqrt{x}\right)^2$$