Exponent and Radical Form Study Guide

Simplest Exponential Form

Laws of Exponents

With rational (fraction) exponents,

let the calculator add, subtract or

multiply. Don't forget MATH FRAC!!

​Going into Simplest Exponential Form

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In Multiplication...

• If the numbers don't have exponents, just multiply

• same base? Just add exponents

• Simplify or Reduce if possible

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​Going into Simplest Exponential Form

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Raising to a power...

• Multiply exponents

• -5 raised to the 2nd

• any # raised to an EVEN exponent is positive

​Going into Simplest Exponential Form

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In Division...

• If numbers don't have exponents, just divide

• same base? subtract exponents

• Simplify or Reduce if possible

Multiple Choice

Simplest Exponential Form: $$\left(-4a^{\frac{3}{4}}\right)^2$$

• $$-16a^{\frac{3}{2}}$$

• $$-8a^{\frac{11}{4}}$$

• $$16a^{\frac{11}{4}}$$

• $$16a^{\frac{3}{2}}$$

Multiple Choice

Simplify: $$\left(4a^{\frac{2}{5}}\right)\cdot\left(3a^{\frac{3}{5}}\right)$$

• $$7a$$

• $$12a$$

• $$7a^{\frac{6}{5}}$$

• $$12a^{\frac{6}{5}}$$

Multiple Choice

Simplify: $$\frac{4^{\frac{7}{3}}a^{\frac{10}{3}}}{4^{\frac{1}{3}}a^{\frac{1}{3}}}$$

• $$a^3$$

• $$8a^3$$

• $$16a^3$$

• $$64a^3$$

Multiple Choice

Simplify 4^1/3 x 4^4/3

• $$4^{\frac{5}{3}}$$

• $$4^{\frac{4}{9}}$$

• $$16^{\frac{5}{3}}$$

Multiple Choice

(6^3/4)^1/3

• 6^1/4
• 6^3/12
• 6^4/7
• 6^9/12

Exponent to Radical Form

See if the number is a perfect factor first. If it is not, go to y = #/x^root

With variables, you can divide the exponent by the root and determine the "leftovers"

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Simplify Radicals

Exponent to Radical Form

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Raising to the 1/2 is the same as square root...

• simplify the # in the radical

• simplify the variable in the radical

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Exponent to Radical Form

Multiple Choice

Simplest Radical Form.

Multiple Choice

Write in Radical form and simplify. $$1024^{\frac{1}{3}}$$  

• 
  $$8\sqrt[3]{2}$$

• $$512\sqrt[3]{2}$$

• $$2\sqrt[3]{8}$$

• $$341\sqrt[3]{x}$$

Multiple Choice

Write in Radical form and simplify: $$\left(81x^{10}\right)^{\frac{1}{4}}$$

• $$9x^5$$

• $$20x^2\sqrt[4]{x^2}$$

• $$3x^2\sqrt[4]{x^2}$$

• $$3x^{\frac{5}{2}}$$

Multiple Choice

Write in Simplest Radical Form.

$$\left(375x^5\right)^{\frac{1}{3}}$$

• $$375x\sqrt[3]{x^2}$$

• $$x\sqrt[3]{375x^2}$$

• $$5x\sqrt[3]{3x^2}$$

• $$\sqrt[3]{375x^{\frac{5}{3}}}$$

Multiple Choice

Write the expression in radical form:
$$7\left(12w\right)^{\frac{1}{2}}$$  

• $$7\sqrt{12w}$$  

• $$\sqrt{84w}$$    

• $$14\sqrt{3w}$$  
  

Radical to Exponential Form

Radical to Exponential Form

• Remember, 729 is a different base than x

• Handle them separately: exponent in numerator and root in denominator

• Since no exponent shows for 729, use 1

Multiple Choice

Write in Exponential Form

• 4^4/3
• 4^3/4
• 2^4/3
• 2^3/4

Multiple Choice

Write in exponential form
√x⁵

• X¹∕ ⁵
• X ²⁄⁵
• X⁵⁄ ²
• X⁵

Multiple Choice

 Write in exponential form
 ∛x⁷

• x⁷∕ ³
• x3⁄7
• 3x⁷
• 7x³

Multiple Choice

Write in Exponential Form and simplify. $$6\sqrt[3]{x^{12}y^5}$$

• $$6x^4y^{\frac{5}{3}}$$  

• $$6x^9y^2$$  

• $$x^2y^{\frac{5}{6}}$$  

Multiple Choice

Write in Exponential form and simplify. $$\sqrt[4]{\left(16a^3\right)^2}$$

• $$16^{\frac{2}{4}}a^{\frac{6}{4}}=4a^{\frac{3}{2}}$$  

• $$16^{\frac{1}{4}}a^{\frac{6}{4}}\ =2a^{\frac{3}{2}}$$  
  

• $$16^{\frac{3}{4}}a^{\frac{5}{4}}$$  

Multiple Select

Select BOTH the Radical form and Exponential form of:
$$\left(9x^2\right)^{\frac{1}{4}}\cdot\left(9x^3\right)^{\frac{1}{4}}$$  

• $$3x^{\frac{5}{4}}$$

•   $$3x\sqrt[4]{x}$$

•   $$3x^{\frac{5}{3}}$$

• $$3x\sqrt[3]{x^2}$$