QUESTION 1: Evaluating an expression with a negative exponent
Rewrite the
following without an exponent: 8^-2
EXPLANATION:
For any nonzero number a and any whole number n,
we have the following.
Rule 1: a^-n
= 1/a^n
Move a^-n to the denominator and change -n to n
Rule 2: 1/a^-n
= a^n
Move a^-n to the numerator and change -n to n
We need to
rewrite 8^-2 without an exponent:
To do this, we first use Rule 1 and move 8^-2 to the denominator, making
the exponent positive and evaluate
8^-2 = 1/8^2 using Rule 1
= 1/64 Since 8^2 =
64
ANSWER: 1/64
QUESTION 2: Introduction to the power of a power rule of exponents
Simplify and
write your answer without parentheses: (z^4)^3
EXPLANATION:
Method 1: By definition,
the exponent 3 tells us how many times z^4 appears in the product.
(z^4)^3 = z^4 * z^4 * z^4
Each exponent of 4 in this
product tells us how many z's to multiply.
(z^4)^3 = z^4 * z^4 * z^4
= z^4*z^4*z^4 * z^4*z^4*z^4 *
z^4*z^4*z^4
We see that we are multiplying a
total of 3 * 4 = 12 z’s. So, we have the following:
(z^4)^3 = z^12
Note that we get 12 by
multiplying the exponents 3 and 4.
Method 2: The method
above suggests a rule called the power of a power rule of exponents.
It says that for any integers m
and n, and any number a, we have the following:
(a^m)^n = a^(m*n)
Using the rule with the current
problem, we get the following:
(z^4)^3 = z^(4*3)
= z^12
ANSWER: z^12
QUESTION 3: Factoring a quadratic with leading coefficient 1
Factor: y^2 – 9y + 20
EXPLANATION:
To factor y^2 – 9y + 20 is to write it as a product (y + m)(y + n) , where m
and n are integers.
y^2 – 9y +
20 = (y + m)(y + n)
From this
equation, we get that the product of m and n is 20
and their sum is -9.
We see that m
= -4 and n = -5 have a product of 20 and a sum of -9. So we get the following:
y^2 – 9y +
20 = (y - 4)(y - 5)
ANSWER: (y -
4)(y - 5)
QUESTION 4: Order of operations with integers
Evaluate 6/(-3)–(-3)
EXPLANATION:
There are two operations in this problem: a division and a subtraction.
We must follow the rules for
order of operations.
We do multiplication and
division before addition and subtraction.
6/(-3)–(-3) = -2 – (-3) Do the division before the subtraction.
= -2 + 3 Change subtraction to addition, change the sign of
the number that was subtracted
= 1
ANSWER: 1
QUESTION 5: Quotient of expressions involving exponents
Simplify: x^5y^4 / xy^6
EXPLANATION:
We simplify as follows
x^5y^4 / xy^6 = x^4y^4 / y^6 Canceling the common factor x
= x^4 / y^2 Canceling the common factor y^4
ANSWER: x^4
/ y^2
QUESTION 6: Exponents and signed fractions
Evaluate and write your answers as fractions: (-1/3)^2, 3 / 4^3
EXPLANATION:
These expressions involve exponents. We
evaluate as follows
(-1/3)^2 = -1/3 * -1/3 = 1/9
3 / 4^3 = 3 / (4*4*4) = 3/64
Note that
the exponent applies only to the denominator
ANSWER:
(-1/3)^2 = 1/9
3 / 4^3 = 3/64
QUESTION 7: Evaluating an expression with a negative exponent
Rewrite the following without an
exponent: (5/2)^-3
EXPLANATION:
For any non-zero rational number a/b and any whole
number n, we have the following
(a/b)^-n = (b/a)^n
= b^n / a^n
We'll apply this rule to
(5/2)^-3
(5/2)^-3 = (2/5)^3
= 2^3 / 5^3
= 8/125
ANSWER:
8/125
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