Rational Exponents and Radical Functions

Radical and Rational Notation

Converting between styles

Radical Notation

• Identify the index

• Identify the radicand

• Identify the exponent on the radicand

Multiple Choice

$$\sqrt{x^4}$$  What is the index of this radical?

• 2

• 4

• x

Multiple Choice

$$\sqrt{x^4}$$  What is the radicand of this radical?

• 2

• x

• 4

Multiple Choice

$$\sqrt{x^4}$$  What is the exponent on the radicand?

• 2

• x

• 4

Rational Notation

Rational notation is a signal that a radical is being used. You can identify rational notation when a base has a fractional exponent. Open the link below in a new window.

https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:exp/x2ec2f6f830c9fb89:rational-exp/v/basic-fractional-exponents

Multiple Choice

$$x^{\frac{3}{4}}$$  What is numerator of the exponent?

• 3/4

• x

• 3

• 4

Multiple Choice

$$x^{\frac{3}{4}}$$  What is the denominator of the exponent?

• 3/4

• x

• 3

• 4

Converting between notation

a is the radicand and the base

m is the exponent and the numerator

n is the index and the denominator

Multiple Choice

$$625^{\frac{3}{4}}=$$  

• $$^4\sqrt{625^3}$$  

• $$^3\sqrt{625^4}$$  

Multiple Choice

$$1000^{\frac{2}{3}}=$$  

• $$^3\sqrt{1000^2}$$  

• $$^2\sqrt{1000^3}$$  

Multiple Choice

$$121^{-\frac{3}{5}}=$$  

• $$\frac{1}{^3\sqrt{121^5}}$$  

• $$\frac{1}{^5\sqrt{121^3}}$$  

• $$^5\sqrt{121^3}$$  

• $$^3\sqrt{121^5}$$  

Multiple Choice

$$^5\sqrt{y^2}=$$  

• $$y^{\frac{2}{5}}$$  

• $$y^{\frac{5}{2}}$$  

Multiple Choice

$$^4\sqrt{\left(mn\right)^3}$$  =

• $$\left(mn\right)^{\frac{4}{3}}$$  

• $$\left(mn\right)^{\frac{3}{4}}$$  

Multiple Choice

$$^{15}\sqrt{15xy}$$  =

• $$\left(15xy\right)^{15}$$  

• $$\left(15xy\right)^{\frac{1}{15}}$$  

Simplifying

Remember your old properties of exponents when simplifying expressions.

Multiple Choice

Simplify

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

• $$^3\sqrt{\frac{x}{3}}$$  

• $$^3\sqrt{\frac{3}{x}}$$  

• $$3\sqrt{\left(\frac{x}{3}\right)^2}$$  

• $$^3\sqrt{\left(\frac{3}{x}\right)^2}$$  

Multiple Choice

$$\left(\frac{x^2}{x^{-11}}\right)^{\frac{1}{3}}$$  =

• $$x^3$$  

• $$^{13}\sqrt{x^3}$$  

• $$^3\sqrt{x^{13}}$$  

• $$^9\sqrt{x^3}$$  

Multiple Choice

Convert to a radical

• y^1/2

• y²

• y³

• y

Multiple Choice

Convert the following to radical form and simplify.

• $$\sqrt{100}$$

• 100

• $$\sqrt{10}$$

• 10

Multiple Choice

When taking the square root of an exponent, you must divide the exponent by 2 and put it outside the square root sign. For example  $$\sqrt{x^{10}}$$  is  $$x^5$$  

• True

• False

Multiple Choice

Simplify the following.

• 1/8

• 8

• -1/8

• -8

Multiple Choice

What is the first step in simplifying this exponent?

• Convert it to a negative radical

• Multiply by -3 and divide by 4

• put it under 1 and make the exponent a positive

• none of the above

Multiple Choice

According to the rules of rational exponents, what would I do to the exponents in this problem?

• add them

• subtract them

• divide them

• multiply them

Multiple Choice

Simplify this radical

• 64^11/2

• 512m⁶

• 64^-6

• none of the above

Multiple Choice

What will I do with the exponents in this example?

• subtract them

• add them

• divide them

• multiply them

Multiple Choice

Simplify this example-leave in exponential form.

• $$2x^{\frac{5}{2}}$$

• $$4x^{\frac{3}{2}}$$

• $$2x^{\frac{3}{2}}$$

• none of the above

Multiple Choice

Anything raised to the zero power equals what???

• zero!

• 1