Download Simplifying Algebraic Expressions Involving Exponents

4.4
Simplifying Algebraic
Expressions Involving
Exponents
GOAL
Simplify algebraic expressions involving powers and radicals.
LEARN ABOUT the Math
The ratio of the surface area to the volume of microorganisms affects their ability
to survive. An organism with a higher surface area-to-volume ratio is more
buoyant and uses less of its own energy to remain near the surface of a liquid,
where food is more plentiful.
Radius (mm)
1
4p
4
a pb
3
1.5
9p
4.5p
2
16p
32
a pb
3
Mike is calculating the surface area-to-volume ratio for different-sized cells. He
assumes that the cells are spherical.
For a sphere,
SA(r) 5 4pr 2
and
V(r) 5 43pr 3.
He substitutes the value of the radius into each formula and then divides the two
expressions to calculate the ratio.
2.5
?
Surface Area/
Volume
25p
125
a
pb
6
3
36 p
36 p
3.5
49p
343
a
b
6p
How can Mike simplify the calculation he uses?
Chapter 4 Exponential Functions
231
EXAMPLE
Simplify
1
Representing the surface area-to-volume ratio
SA(r)
, given that SA(r) 5 4pr 2 and V(r) 5 43 pr 3.
V(r)
Bram’s Solution
SA(r)
V(r)
I used the formulas for SA and V
and wrote the ratio.
The numerator and denominator
have a factor of p, so I divided
both by p.
I started to simplify the expression
by dividing the coefficients.
1
4pr 2
5
4 3
pr
31
5 3r 21
5
3
r
a4 4
3
4
5 4 3 5 3b
3
4
The bases of the powers were the
same, so I subtracted exponents to
simplify the part of the expression
involving r.
I used a calculator and substituted
r 5 2 in the unsimplified ratio first
and my simplified expression next.
Each version gave me the same
answer, so I think that they are
equivalent, but the second one
took far fewer keystrokes!
Reflecting
232
A.
How can you use the simplified ratio to explain why the values in Mike’s
table kept decreasing?
B.
Is it necessary to simplify an algebraic expression before you substitute
numbers and perform calculations? Explain.
C.
What are the advantages and disadvantages to simplifying an algebraic
expression prior to performing calculations?
D.
Do the exponent rules used on algebraic expressions work the same way as
they do on numerical expressions? Explain by referring to Bram’s work.
4.4 Simplifying Algebraic Expressions Involving Exponents
APPLY the Math
EXAMPLE
Simplify
2
Connecting the exponent rules to the
simplification of algebraic expressions
(2x23y 2 ) 3
.
(x 3 y24 )2
Adnan’s Solution
(2 x23y 2 ) 3
(2) 3 (x23 ) 3 ( y 2 ) 3
5
(x 3y24 ) 2
(x 3 ) 2 ( y24 ) 2
5
8x29y 6
x 6y28
5 8x2926y62 (28)
5 8x215y14
EXAMPLE
8y
x15
3
I simplified the whole expression by
subtracting exponents of terms with
the same base.
One of the powers had a negative
exponent. To write it with positive
exponents, I used its reciprocal.
14
5
I used the product-of-powers rule to
raise each factor in the numerator
to the third power and to square
each factor in the denominator.
Then I multiplied exponents.
Selecting a computational strategy
to evaluate an expression
Evaluate the expression
(x 2n11 ) (x 3n21 )
for x 5 23 and n 5 2.
x 2n25
Bonnie’s Solution: Substituting, then Simplifying
(x 2n11 ) (x 3n21 )
(23) 2(2) 11 (23) 3(2) 21
5
x 2n25
(23) 2(2) 25
5
5
(23) 5 (23) 5
(23) 21
(2243) (2243)
1
23
I substituted the values for x
and n into the expression.
Then I evaluated the
numerator and denominator
separately, before dividing
one by the other.
5 2177 147
Chapter 4 Exponential Functions
233
Alana’s Solution: Simplifying, then Substituting
Each power had the same base, so
I simplified by using exponent rules
before I substituted.
(x 2n11 ) (x 3n21 )
x 2n25
5
5
x (2n11) 1 (3n21)
x 2n25
I added the exponents in the
numerator to express it as a single
power.
x 5n
x 2n25
Then I subtracted the exponents in
the denominator to divide the
powers.
5 x(5n) 2 (2n25)
Once I had a single power,
I substituted 23 for x and 2 for n
and evaluated.
5 x 3n15
5 (23) 3(2) 15
5 (23) 11
5 2177 147
EXAMPLE
4
Simplifying an expression involving powers with
rational exponents
1
Simplify
(27a 23b 12 ) 3
1
.
(16a 28b 12 ) 2
Jane’s Solution
1
(27a23b12 ) 3
1
28 12 2
(16a b )
1 23 12
3 3
3
5
27 a b
1 28 12
162a 2 b 2
3a21b4
5 24 6
4a b
3
5 a2114b426
4
3
5 a 3b22
4
5
234
3a 3
4b 2
4.4 Simplifying Algebraic Expressions Involving Exponents
In the numerator, I applied the
1
exponent 3 to each number or
variable inside the parentheses,
using the power-of-a-power rule.
I did the same in the denominator,
1
applying the exponent 2 to the
numbers and variables.
I simplified by subtracting the
exponents.
I expressed the answer with positive
exponents.
Sometimes it is necessary to express an expression involving radicals using
exponents in order to simplify it.
EXAMPLE
Simplify a
5
Representing an expression involving radicals
as a single power
5 8 3
!
x
b.
!x 3
Albino’s Solution
5 8 3
!
x5 3
x
a 3 b 5 a 3b
!x
x2
8
8
5 (x5
2
16
5 (x10
Since this is a fifth root divided by a
square root, I couldn’t write it as a
single radical.
3
2 3
2
I changed the radical expressions to
exponential form and used exponent
rules to simplify.
)
15
10 3
)
1
10 3
5 (x )
3
When I got a single power, I
converted it to radical form.
5 x10
5 !x 3
10
In Summary
Key Idea
• Algebraic expressions involving powers containing integer and rational
exponents can be simplified with the use of the exponent rules in the same
way numerical expressions can be simplified.
Need to Know
• When evaluating an algebraic expression by substitution, simplify prior to
substituting. The answer will be the same if substitution is done prior to
simplifying, but the number of calculations will be reduced.
• Algebraic expressions involving radicals can often be simplified by changing
the expression into exponential form and applying the rules for exponents.
CHECK Your Understanding
1. Simplify. Express each answer with positive exponents.
a)
x 4 (x 3 )
b)
( p23 ) ( p) 5
m5
m23
a24
d) 22
a
c)
e) ( y 3 ) 2
f)
(k 6 ) 22
Chapter 4 Exponential Functions
235
2. Simplify. Express each answer with positive exponents.
a)
y 10 ( y 4 ) 23
c)
(n 24 ) 3
(n 23 ) 24
e)
(x21 ) 4x
x23
b)
(x23 ) 23 (x21 ) 5
d)
w 4 (w23 )
(w 22 ) 21
f)
(b27 ) 2
b(b25 )b9
x 7 ( y 2) 3
.
x 5y 4
a) Substitute x 5 22 and y 5 3 into the expression, and evaluate it.
b) Simplify the expression. Then substitute the values for x and y to
evaluate it.
c) Which method seems more efficient?
3. Consider the expression
PRACTISING
4. Simplify. Express answers with positive exponents.
a)
(pq 2 ) 21 (p 3q 3 )
b)
a
x 3 22
b
y
(w 2x) 2
(x 21 ) 2w 3
c)
(ab) 22
b5
e)
d)
m 2n 2
(m 3n 22 ) 2
f) a
(ab) 21 22
b
a 2b 23
5. Simplify. Express answers with positive exponents.
a)
(3xy 4 ) 2 (2x 2y) 3
c)
(10x) 21y 3
15x 3y23
e)
p25 (r 3 ) 2
(p 2r) 2 (p 21 ) 2
b)
(2a3 ) 2
4ab2
d)
(3m 4n 2 ) 2
12m22n 6
f) a
(x 3y) 21 (x 4y 3 ) 21
b
(x 2y 23 ) 22
e) a
(32x 5 ) 22 0.2
b
(x 21 ) 10
6. Simplify. Express answers with positive exponents.
K
1
2
a)
(x 4 ) 2 (x 6 )
b)
9(c 8 ) 0.5
(16c 12 ) 0.25
1
3
c)
d)
"25m212
"36m10
(10x 3 ) 2
Ä (10x 6 ) 21
3
"1024x 20
10
f)
"512x 27
9
7. Evaluate each expression. Express answers in rational form with positive
exponents. 1
a) (16x 6y 4 ) 2 for x 5 2, y 5 1
1
b)
(9p22 ) 2
for p 5 3
6p 2
1
c)
(81x 4y 6 ) 2
1
8(x 9y 3 ) 3
for x 5 10, y 5 5
1
(25a4 ) 21 2
d) a
b for a 5 11, b 5 10
(7a22b) 2
236
4.4 Simplifying Algebraic Expressions Involving Exponents
8. Evaluate. Express answers in rational form with positive exponents.
3
a)
("10 000x) 2 for x 5 16
b)
a
(4x 3 ) 4 20.5
b
for x 5 5
(x 3 ) 6
c) (22a 2b) 23 " 25a 4b 6 for a 5 1, b 5 2
d)
(18m 25n 2 ) (32m 2n) for m 5 10, n 5 1
Ä
4mn 23
9. Simplify. Express answers in rational form with positive exponents.
!64a12 3
c) a 1.5 26 b
(a )
2
a)
(36m4n6 ) 0.5 (81m12n8 ) 0.25
(6x 3 ) 2 (6y 3 )
b
b) a
(9xy) 6
2
1
3
21
(x 18 ) 6 0.5
b
d) a 5
!243x 10
1
(16x 8y 24 ) 4
10. If M 5
, determine values for x and y so that
32x 22 y 8
T
a) M 5 1
b) M . 1
c) 0 , M , 1 d) M , 0
11. The volume and surface area of a cylinder are given, respectively, by the
A
formulas
V 5 pr 2h
and
SA 5 2prh 1 2pr 2.
Determine an expression, in simplified form, that represents the surface
area-to-volume ratio for a cylinder.
b) Calculate the ratio for a radius of 0.8 cm and a height of 12 cm.
a)
12. If x 5 22 and y 5 3, write the three expressions in order from least to
greatest.
y 24 (x 2 ) 23y 23 x 23 ( y 21 ) 22
,
, ( y 25 ) (x 5 ) 22 ( y 2 ) (x 23 ) 24
x 25 ( y 24 ) 2
(x 25 ) ( y 4 )
13. How is simplifying algebraic expressions like simplifying numerical ones?
C
How is it different?
Extending
14. a)
The formula for the volume of a sphere of radius r is V(r) 5 43 p r 3.
Solve this equation for r. Write two versions, one in radical form and one
in exponential form.
3
b) Determine the radius of a sphere with a volume of 256p
3 m .
15. Simplify
! x(x 2n11 )
, x . 0.
3 3n
!
x
Chapter 4 Exponential Functions
237