ElementaryAlgebra2e-WEB_3zxfu3Z-214

Prévia do material em texto

471. ⓐ ⓑ 472. ⓐ ⓑ 473. ⓐ ⓑ
474. ⓐ ⓑ 475. ⓐ ⓑ 476. ⓐ ⓑ
477. ⓐ ⓑ
Use the Quotient Property to Simplify Expressions with Higher Roots
In the following exercises, simplify.
478. ⓐ ⓑ 479. ⓐ ⓑ 480. ⓐ ⓑ
481. ⓐ ⓑ 482. ⓐ ⓑ 483. ⓐ ⓑ
484. ⓐ ⓑ 485. ⓐ ⓑ 486. ⓐ ⓑ
487. ⓐ ⓑ 488. ⓐ ⓑ 489. ⓐ ⓑ
Add and Subtract Higher Roots
In the following exercises, simplify.
490. ⓐ ⓑ 491. ⓐ ⓑ 492. ⓐ ⓑ
493. ⓐ
ⓑ
494. ⓐ ⓑ 495. ⓐ ⓑ
496. ⓐ ⓑ 497. ⓐ ⓑ 498. ⓐ
ⓑ
499. ⓐ
ⓑ
Mixed Practice
In the following exercises, simplify.
500. 501. 502.
503. 504. 505.
506. 507. 508.
509. 510. 511.
512. 513. 514.
1056 9 • Roots and Radicals
Access for free at openstax.org
515. 516. 517.
518. 519.
Everyday Math
520. Population growth The expression
models the growth of a mold population after
generations. There were 10 spores at the start,
and each had offspring. So is the
number of offspring at the fifth generation. At
the fifth generation there were 10,240 offspring.
Simplify the expression to determine
the number of offspring of each spore.
521. Spread of a virus The expression models
the spread of a virus after cycles. There were
three people originally infected with the virus,
and each of them infected people. So is
the number of people infected on the fourth
cycle. At the fourth cycle 1875 people were
infected. Simplify the expression to
determine the number of people each person
infected.
Writing Exercises
522. Explain how you know that . 523. Explain why is not a real number but
is.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
9.8 Rational Exponents
Learning Objectives
By the end of this section, you will be able to:
Simplify expressions with
Simplify expressions with
Use the Laws of Exponents to simply expressions with rational exponents
BE PREPARED 9.21 Before you get started, take this readiness quiz.
Add: .
If you missed this problem, review Example 1.81.
BE PREPARED 9.22 Simplify: .
If you missed this problem, review Example 6.24.
BE PREPARED 9.23 Simplify: .
If you missed this problem, review Example 6.89.
9.8 • Rational Exponents 1057
Simplify Expressions with
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 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 . We will use the Power Property of Exponents to find the value
of p.
Multiply the exponents on the left.
Write the exponent 1 on the right.
The exponents must be equal.
Solve for
But we know also . Then it must be that .
But we know also . Then it must be that .
This same logic can be used for any positive integer exponent n to show that .
Rational Exponent
If is a real number and , .
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.
EXAMPLE 9.103
Write as a radical expression: ⓐ ⓑ ⓒ .
1058 9 • Roots and Radicals
Access for free at openstax.org
Solution
We want to write each expression in the form .
ⓐ
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.
ⓑ
The denominator of the exponent is 3, so the index is 3.
ⓒ
The denominator of the exponent is 4, so the index is 4.
TRY IT 9.205 Write as a radical expression: ⓐ ⓑ ⓒ .
TRY IT 9.206 Write as a radial expression: ⓐ ⓑ ⓒ .
EXAMPLE 9.104
Write with a rational exponent: ⓐ ⓑ ⓒ .
Solution
We want to write each radical in the form .
ⓐ
No index is shown, so it is 2.
The denominator of the exponent will be 2.
ⓑ
The index is 3, so the denominator of the exponent is 3.
9.8 • Rational Exponents 1059
ⓒ
The index is 4, so the denominator of the exponent is 4.
TRY IT 9.207 Write with a rational exponent: ⓐ ⓑ ⓒ .
TRY IT 9.208 Write with a rational exponent: ⓐ ⓑ ⓒ .
EXAMPLE 9.105
Write with a rational exponent: ⓐ ⓑ ⓒ .
Solution
We want to write each radical in the form .
ⓐ
No index is shown, so it is 2.
The denominator of the exponent will be 2.
ⓑ
The index is 3, so the denominator of the exponent is 3.
ⓒ
The index is 4, so the denominator of the exponent is 4.
TRY IT 9.209 Write with a rational exponent: ⓐ ⓑ ⓒ .
TRY IT 9.210 Write with a rational exponent: ⓐ ⓑ ⓒ .
In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.
1060 9 • Roots and Radicals
Access for free at openstax.org
	Chapter 9 Roots and Radicals
	9.8 Rational Exponents