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By the end of this section, you will be able to:
  • Simplify expressions with a 1 n
  • Simplify expressions with a m n
  • Use the Laws of Exponents to simply expressions with rational exponents

Before you get started, take this readiness quiz.

  1. Add: 7 15 + 5 12 .
    If you missed this problem, review [link] .
  2. Simplify: ( 4 x 2 y 5 ) 3 .
    If you missed this problem, review [link] .
  3. Simplify: 5 −3 .
    If you missed this problem, review [link] .

Simplify expressions with a 1 n

Rational exponents are another way of writing expressions with radicals. When we use rational exponents    , we can apply the properties of exponents to simplify expressions.

The Power Property for Exponents says that ( a m ) n = a m · n when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.

Suppose we want to find a number p such that ( 8 p ) 3 = 8 . We will use the Power Property of Exponents to find the value of p .

( 8 p ) 3 = 8 Multiply the exponents on the left. 8 3 p = 8 Write the exponent 1 on the right. 8 3 p = 8 1 The exponents must be equal. 3 p = 1 Solve for p . p = 1 3 So ( 8 1 3 ) 3 = 8 .

But we know also ( 8 3 ) 3 = 8 . Then it must be that 8 1 3 = 8 3 .

This same logic can be used for any positive integer exponent n to show that a 1 n = a n .

Rational exponent a 1 n

If a n is a real number and n 2 , a 1 n = a n .

There will be times when working with expressions will be easier if you use rational exponents    and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.

Write as a radical expression: x 1 2 y 1 3 z 1 4 .

Solution

We want to write each expression in the form a n .


x 1 2 The denominator of the exponent is 2, so the index of the radical is 2. We do not show the index when it is 2. x


y 1 3 The denominator of the exponent is 3, so the index is 3. y 3


z 1 4 The denominator of the exponent is 4, so the index is 4. z 4

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Write as a radical expression: t 1 2 m 1 3 r 1 4 .

t m 3 r 4

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Write as a radial expression: b 1 2 z 1 3 p 1 4 .

b z 3 p 4

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Write with a rational exponent: x y 3 z 4 .

Solution

We want to write each radical in the form a 1 n .


x No index is shown, so it is 2. The denominator of the exponent will be 2. x 1 2


y 3 The index is 3, so the denominator of the exponent is 3. y 1 3


z 4 The index is 4, so the denominator of the exponent is 4. z 1 4

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Write with a rational exponent: s x 3 b 4 .

s 1 2 x 1 3 b 1 4

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Write with a rational exponent: v p 3 p 4 .

v 1 2 p 1 3 p 1 4

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Write with a rational exponent: 5 y 4 x 3 3 5 z 4 .

Solution

We want to write each radical in the form a 1 n .


  1. 5 y No index is shown, so it is 2. The denominator of the exponent will be 2. ( 5 y ) 1 2


  2. 4 x 3 The index is 3, so the denominator of the exponent is 3. ( 4 x ) 1 3


  3. 3 5 z 4 The index is 4, so the denominator of the exponent is 4. 3 ( 5 z ) 1 4
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Write with a rational exponent: 10 m 3 n 5 3 6 y 4 .

( 10 m ) 1 2 ( 3 n ) 1 5 ( 486 y ) 1 4

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Write with a rational exponent: 3 k 7 5 j 4 8 2 a 3 .

( 3 k ) 1 7 ( 5 j ) 1 4 ( 1024 a ) 1 3

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In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.

Simplify: 25 1 2 64 1 3 256 1 4 .

Solution


25 1 2 Rewrite as a square root. 25 Simplify. 5


64 1 3 Rewrite as a cube root. 64 3 Recognize 64 is a perfect cube. 4 3 3 Simplify. 4


256 1 4 Rewrite as a fourth root. 256 4 Recognize 256 is a perfect fourth power. 4 4 4 Simplify. 4

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Questions & Answers

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When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
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Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
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Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
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Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
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In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
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In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .
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Answer
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c
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the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
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suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product
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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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