Transcript 7-1 Integer Exponents

7-1 Integer Exponents
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Warm Up
California Standards
Lesson Presentation
7-1 Integer Exponents
Warm Up
Evaluate each expression for the given
values of the variables.
1. x3y2 for x = –1 and y = 10
2.
–100
for x = 4 and y = (–7)
Write each number as a power of the
given base.
3. 64; base 4 43
4. –27; base (–3) (–3)3
7-1 Integer Exponents
California
Standards
2.0 Students understand and use such
operations as taking the opposite, finding the
reciprocal, taking a root, and raising to a fractional
power. They understand and use the rules of
exponents.
7-1 Integer Exponents
You have seen positive exponents. Recall that
to simplify 32, use 3 as a factor 2 times:
32 = 3  3 = 9.
But what does it mean for an exponent to be
negative or 0? You can use a table and look for a
pattern to make a conjecture.
Power 55 54 53 52
Value 3125 625 125 25
5
5
5
5
51
5
50
5–1
5–2
7-1 Integer Exponents
When the exponent decreases by one, the value
of the power is divided by 5. Continue the
pattern of dividing by 5.
7-1 Integer Exponents
Remember!
Exponent
x
Base
4
7-1 Integer Exponents
7-1 Integer Exponents
Notice the phrase “nonzero number” in the
previous table. This is because 00 and 0 raised to
a negative power are both undefined. For
example, if you use the pattern given above in
the table with a base of 0 instead of 5, you
would get 00 =
. Also, 0–6 would be
=
.
Since division by 0 is undefined, neither value
exists.
7-1 Integer Exponents
Reading Math
2–4 is read “2 to the negative fourth power.”
7-1 Integer Exponents
Additional Example 1: Measurement Application
One cup is 2–4 gallons. Simplify this expression.
gal is equal to
7-1 Integer Exponents
Check It Out! Example 1
A sand fly may have a wingspan up to 5–3 m.
Simplify this expression.
5-3 m is equal to
7-1 Integer Exponents
Additional Example 2: Zero and Negative Exponents
Simplify.
A. 4–3
B. 70
7º = 1
C. (–5)–4
D. –5–4
Any nonzero number raised to the zero
power is 1.
7-1 Integer Exponents
Caution
In (–3)–4, the base is negative because the
negative sign is inside the parentheses. In –3–4
the base (3) is positive.
7-1 Integer Exponents
Check It Out! Example 2
Simplify.
a. 10–4
b. (–2)–4
c. (–2)–5
d. –2–5
7-1 Integer Exponents
Additional Example 3A: Evaluating Expressions
with Zero and Negative Exponents
Evaluate the expression for the given value of
the variable.
x–2 for x = 4
Substitute 4 for x.
Use the definition
7-1 Integer Exponents
Additional Example 3B: Evaluating Expressions with
Zero and Negative Exponents
Evaluate the expression for the given values of
the variables.
–2a0b-4 for a = 5 and b = –3
Substitute 5 for a and –3 for b.
Evaluate expressions with
exponents.
Write the power in the
denominator as a product.
Evaluate the powers in
the product.
Simplify.
7-1 Integer Exponents
Check It Out! Example 3a
Evaluate the expression for the given value of
the variable.
p–3 for p = 4
Substitute 4 for p.
Evaluate exponent.
Write the power in the
denominator as a product.
Evaluate the powers in
the product.
7-1 Integer Exponents
Check It Out! Example 3b
Evaluate the expression for the given values
of the variables.
for a = –2 and b = 6
Substitute –2 for a and 6 for b.
Evaluate expressions with
exponents.
2
Write the power in the
denominator as a product.
Evaluate the powers in
the product.
Simplify.
7-1 Integer Exponents
What if you have an expression with a negative
exponent in a denominator, such as
?
or
Definition of a
negative exponent.
Substitute –8 for n.
Simplify the exponent
on the right side.
If a base with a negative exponent is in a
denominator, it is equivalent to the same base with
the opposite (positive) exponent in the numerator.
7-1 Integer Exponents
What if you have an expression with a negative
exponent in a denominator, such as
?
or
Definition of a
negative exponent.
Substitute –8 for n.
Simplify the exponent
on the right side.
An expression that contains negative or zero
exponents is not considered to be simplified.
Expressions should be rewritten with only positive
exponents.
7-1 Integer Exponents
Additional Example 4: Simplifying Expressions with
Zero and Negative Numbers
Simplify.
A. 7w–4
B.
7-1 Integer Exponents
Additional Example 4: Simplifying Expressions with
Zero and Negative Numbers
Simplify.
C.
and
7-1 Integer Exponents
Simplify.
Check It Out! Example 4
a. 2r0m–3
r0 = 1 and
b.
c.
.
7-1 Integer Exponents
Lesson Quiz: Part I
1. A square foot is 3–2 square yards. Simplify this
expression.
Simplify.
2. 2–6
3. (–7)–3
4. 60
5. –112
1
–121
7-1 Integer Exponents
Lesson Quiz: Part II
Evaluate each expression for the given
value(s) of the variables(s).
6. x–4 for x =10
7.
for a = 6 and b = 3
Simplify.
8. 4y–5
10.
9.
–3y6